direct reconstruction of the quantum density operator via

Universita degli Studi di Padova

Dipartimento di Fisica e Astronomia “G. Galilei”

Dipartimento di Ingegneria dell’Informazione

Corso di Laurea Magistrale in Fisica

Direct Reconstruction of the QuantumDensity Operator via Measurements of

Arbitrary Strength

Author:Giulio Foletto

Supervisor:Prof. Giuseppe Vallone

September 2017

http://www.unipd.it

http://www.dfa.unipd.it/

https://www.dei.unipd.it/

UNIVERSITA DEGLI STUDI DI PADOVA

Dipartimento di Fisica e Astronomia “G. Galilei”Dipartimento di Ingegneria dell’Informazione

Corso di Laurea Magistrale in Fisica

Direct Reconstruction of the Quantum Density Operator via Measurementsof Arbitrary Strength

by Giulio Foletto

Abstract

One of the fundamental problems of experimental quantum physics is the determinationof the state of a system, which provides its accurate description and contains all theavailable information about it. The standard solution is quantum state tomography, atechnique that employs a large number of measurements of complementary observablesto reconstruct a general density operator. Recently, new protocols have been proposedwhich allow to obtain the matrix elements one by one with fewer and less complicatedobservations by exploiting so-called weak measurements, that perturb the system lessthan ideal projections at the cost of being less precise.

This thesis presents an evolution of these schemes that eschews the weakness of themeasurement, thus gaining in the accuracy of the results and in practical feasibility. Werealized an experiment that uses these methods to reconstruct the density operator ofthe polarization state of light in single photons regime.

Acknowledgements

I would like to thank first my supervisor Prof. Giuseppe “Pino” Vallone for having givenme the opportunity to work on this project and for having provided direction and muchneeded help. Not only he made this work possible, that would be true of any thesissupervisor, but he made it enjoyable with his passion and energy. He and Prof. PaoloVilloresi are at the helm of a top notch research group in one of the most promisingfields of physics and technology, but their merit is also the creation of a great team ofpeople with whom my collaboration has been easy and fun.

On this note I would also like to thank everyone working at the LUXOR laboratory forthe help and for having accepted me as part of the family. In particular, Luca Calderarohas been fundamental for the experimental part of this thesis and I thank him for havingintroduced me to the lab life and for having taught me most of what I now know aboutthe technical aspects of quantum optics.

Finally I thank my family, friends and girlfriend for having put up with me in the lastfew months, during which most of my replies were “Sorry, I’m at the lab right now” or“Back to writing my thesis, bye!”. Despite my love for physics, without you this workand the five years before it would have been much more difficult.

ii

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http://www.dfa.unipd.it/

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Contents

Abstract ii

Acknowledgements ii

Table of Contents iii

1 Introduction 1

1.1 Mixed States and Density Operators . . . . . . . . . . . . . . . . . . . . . 2

1.2 Purification of a Mixed State . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The Bidimensional Case: The Qubit . . . . . . . . . . . . . . . . . . . . . 5

1.4 Quantum State Tomography . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Weak Measurements 9

2.1 Weak Values and the Ancilla Measurement Scheme . . . . . . . . . . . . . 9

2.2 The Qubit Pointer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Applications of the Weak Value . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Amplification through Postselection . . . . . . . . . . . . . . . . . 13

2.3.2 The Meaning of the Weak Value and the Quantum Three-BoxParadox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 Average Trajectories of Photons in a Double Slit Experiment . . . 15

2.3.4 Monogamy of Bell’s Inequalities Violations . . . . . . . . . . . . . 16

2.4 Direct Weak Measurement of a Pure Quantum State . . . . . . . . . . . . 16

2.5 Direct Weak Measurement of a General Quantum State . . . . . . . . . . 18

2.5.1 Direct Weak Reconstruction of the Density Operator . . . . . . . . 19

2.5.2 Direct Weak Reconstruction of the Dirac Distribution . . . . . . . 20

2.6 Normalization of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Direct State Reconstruction via Strong Measurements 23

3.1 Evolution of the Basic Protocol to Couplings of Arbitrary Strength . . . . 23

3.2 Evolution of the Advanced Protocols . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 The Density Operator Protocol . . . . . . . . . . . . . . . . . . . . 24

3.2.2 The Dirac Distribution Protocol . . . . . . . . . . . . . . . . . . . 25

3.2.3 A Summary of the two Protocols . . . . . . . . . . . . . . . . . . . 26

3.3 Comparison with Weak Reconstruction Protocols . . . . . . . . . . . . . . 27

3.3.1 The Inherent Bias of Weak Measurements . . . . . . . . . . . . . . 27

3.3.2 The Role of Experimental Errors . . . . . . . . . . . . . . . . . . . 28

3.4 Comparison with QST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Measuring Products of Non-Commuting Observables with Strong Couplings 32

iii

iv

4 The Experiment 35

4.1 The Source of Entangled Photons . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Correlation: The SPDC process . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Superposition: The Sagnac Interferometer . . . . . . . . . . . . . . 38

4.1.3 Performance and Polarization Stability . . . . . . . . . . . . . . . . 40

4.1.4 Temporal Coherence of the Photons . . . . . . . . . . . . . . . . . 41

4.2 The Measurement System . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.2 Practical Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.3 Phase Calibration and Stability . . . . . . . . . . . . . . . . . . . . 48

4.2.4 Analysis of the Statistical Error . . . . . . . . . . . . . . . . . . . . 50

4.3 Weakening the Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Results 55

5.1 Results at Maximum Strength . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Results for Variable Strength . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Conclusions 65

A Proof of the Theoretical Results 67

A.1 Measuring the Weak Value . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.2 Weak Measurements of Products . . . . . . . . . . . . . . . . . . . . . . . 70

A.3 Derivation of the Measurement Protocols . . . . . . . . . . . . . . . . . . 74

A.4 Strong Measurements of Products . . . . . . . . . . . . . . . . . . . . . . 81

A.5 Details about the Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.5.1 Generation of Random Density Matrices . . . . . . . . . . . . . . . 83

A.5.2 Simulation of the Experiment . . . . . . . . . . . . . . . . . . . . . 84

Bibliography 85

Chapter 1

Introduction

At the very foundation of every physical theory stands the concept of state. A state isoften a mathematical entity that, together with a physical interpretation, can describea system and allows the determination of (at least some of) its properties. It does soby restricting such properties “by as many conditions or data as theoretically possible”[1, p. 11]. For instance, thermodynamics can define the state of an ideal gas with thevariables P (pressure), V (volume) and T (absolute temperature), whereas in classicalmechanics the phase space of positions and momenta is the domain of the possible statesof a particle. Given perfect knowledge of a state and of the laws governing its evolution(e.g. a sequence of transformations of the gas, or the Hamilton’s equations of motion),many theories can find the entire future and past of a system.

Quantum physics is by nature not so deterministic. On account of the superposition prin-ciple, its states reside in a vector structure, precisely a Hilbert space [2, Ch. 2, Sec. 1], butare described by rays so that, intuitively, the sum of a state with itself does not changeit. Unit vectors are conventionally chosen as representatives of such rays and containall the information that can ideally be extracted from the particular configuration thesystem is in. Even with this knowledge, only the outcomes of some measurements canbe predicted for sure, while others remain aleatory because of the uncertainty principle.

Moreover, a distinction has to be made between these cases (pure states) and the onesin which even less information is available (mixed ones). For the latter situations, moregeneral descriptions such as that of density operators, must be introduced. Just as withrays in a Hilbert space, with this formalism it is possible to calculate the mean valuesand all other statistical moments of any measurement from a single mathematical object.

But then how does one find it? Quantum state tomography (QST) comes to the aid.It is a set of techniques that aim to reconstruct the state of a system by means ofperforming specific measurements on many identical copies of it. It has been usedextensively for the purpose of characterizing devices that produce quantum systems ina predetermined state, but its importance has recently been growing in the promisingfields of quantum information and communication. Here, states are used to encode andtransmit a message, so that their precise knowledge can become relevant for the success ofan exchange or the certification of the involved parties. Moreover, the quantum computerof the future will need verification of the states it uses as inputs for its calculations andvalidation for the reading of its results. In more fundamental research, QST can be usedas the final step of quantum simulations to find and characterize the ground state ofcomplicated Hamiltonians.

1

2 Introduction

Up until recently, QST of d-dimensional states has involved O(d2) measurements and alinear reconstruction of the entire density operator. Yet, in 2011 Lundeen et al. proposeda scheme that can directly obtain each single element of a quantum wavefunction [3] ordensity operator [4] in a fixed number of measurements and a few simple calculationsthat in general do not depend on the nature of the system to be studied or the dimensiond of its associated Hilbert space. However, this protocol is based on the concepts of weakvalues and weak couplings between the system and an ancilla, the measurement devicethat actually provides the needed information, but new theoretical work suggests thatthis weakness is not necessary and even counter-productive for tomography [5].

The main goal of this thesis is to expand on these ideas and show that the densityoperator can be directly measured within the framework of ancillae but without theuse of weak couplings between systems and with better results in terms of accuracyand precision. After a brief review of the necessary formalism and of the proceduresof standard QST, I will explain the fundamentals of weak values and measurementsand their use in state reconstruction schemes. I shall then propose a new protocolbased on strong couplings and present an experimental realization aimed at finding thepolarization of a photon system.

1.1 Mixed States and Density Operators

In order for this thesis to be as self-contained as possible, the rest of this chapter willprovide the rudiments of the formalism that will be used in the following dissertation.However, I shall not detail the mathematics of pure states more than I have alreadydone and I will assume that the reader is familiar with the very basic principles ofquantum mechanics, the Dirac notation and concepts such as observables, projectors,unitary evolutions, mean values, ideal measurements, uncertainty and entanglement [6].Yet, since the quantum state is clearly fundamental for this work, I will focus on how itis generalized beyond the restrictions of purity and what are the main ways in which itis usually represented.

No realistic process is so accurate that it can prepare perfectly pure states, therefore allthe quantum systems that can be encountered in practice should be considered statisticalmixtures, better known as mixed states. In order to define them, consider the situationin which one knows that the system under study can be represented by one of n possiblekets (rays) |φ1〉..|φn〉, each with an associated probability p1..pn such that

∑nk=1 = 1.

The most common description of this configuration is offered by the density operator

% ≡n∑k=1

pk|φk〉〈φk| (1.1)

which can be seen as a complex d × d matrix if d is the dimension of the Hilbertspace of the system. The presence of a bra 〈φk| for each corresponding ket |φk〉 makesthis expression manifestly invariant for phase multiplication, another advantage of thisformalism over that of Hilbert space vectors.

Its other fundamental properties are:

• % is hermitian: % = %†. Consequently, like any observable, it can be spectrally de-composed and expressed as a real combination of projectors %′ =

∑j λj

∑r |λj,r〉〈λj,r|.

Introduction 3

Although % and %′ apparently refer to different mixtures, they represent the samemixed state.

• % is positive semidefinite: 〈ψ|%|ψ〉 ≥ 0 ∀ψ or equivalently λj ≥ 0 ∀j.

• % has unit trace: Tr(%) = 1. This condition is equivalent to the normalizationrequest for vectors that describe pure states and is deeply integrated into theformalism.

It is clear that, as a consequence of these characteristics, Tr(%2) ≤ 1. This quantity iscalled the purity of the system and is equal to 1 if and only if the state described by %is pure, whereas it takes its minimum value d−1 in the completely mixed state. Noticethat % has d2 − 1 real degrees of freedom.

All the rules of quantum mechanics that are defined in the ket language can easily beextended to density operators. For example the evolution of a state according to unitarytransformation U is

%→n∑k=1

pkU |φk〉〈φk|U † = U%U † (1.2)

whereas the m-th statistical moment of the measurement of observable A is simply

〈Am〉% =n∑k=1

pk〈Am〉φk =n∑k=1

pk〈φk|Am|φk〉 = Tr(%Am) (1.3)

This is reminiscent of the Born rule, according to which the probability of obtaining thej-th outcome (i.e. the j-th eigenvalue) of the measurement of A is

Prj =∑k

pk‖Πj |φk〉‖2 = Tr(%Πj) (1.4)

and if this happens, the state collapses onto

Πj %Πj

Prj(1.5)

where the denominator is needed for renormalization of the trace. In what follows, I willmostly use the density operator formalism for the important results, and revert to unitvectors when simplicity is a priority.

There are also other mathematical entities that contain the same information of %, suchas the Dirac distribution, which is of tangential interest for this work [7, 8]. Originallya way to represent operators using a set of non-commuting observables, it can be seen

as the Fourier transform of the density matrix. Indeed if {|bl〉 ≡ 1√d

∑j |aj〉e

i 2πjld | l =

0..d− 1} is the Fourier basis of {|aj〉 | j = 0..d− 1}, element jl of the (discrete) Diracdistribution that describes the same state of % coincides with

Djl = 〈aj |%|bl〉〈bl|aj〉 =1

d

d−1∑k=0

〈aj |%|ak〉ei2πl(k−j)

d (1.6)

Other descriptions for quantum states are used in specific fields [9] such as the Wignerdistribution [10, 11] or the Moyal M-Function [12] in statistical mechanics, the Husimi

4 Introduction

Q-Function [13] or the Glauber-Sudarshan P-Representation [14, 15] in quantum optics.Formulations like these are rarely used because they do not share all the properties of agood probability distribution. Nonetheless, in some cases they can be an intermediatestep to retrieve the density operator, as in one of the protocols that I shall detail in thenext chapter, or in other methods [16, 17].

1.2 Purification of a Mixed State

In order to better understand the nature of mixed states, some new mathematical toolsmust be introduced. Given a tensor product Hilbert space HA ⊗ HB and a densityoperator % on it, the partial trace of % on HB is defined as:

TrHB(%) ≡

dB∑l=1

〈bl|%|bl〉 = %A (1.7)

where {|bl〉} is a basis for HB. Any information that can be extracted by focusing onlyon subsystem HA is contained in %A. This means that the mean value of any observablethat acts only on HA (and therefore is written as an operator on HA ⊗HB as A ⊗ 1)can be calculated by ignoring (tracing out) HB:

〈A⊗ 1〉% = Tr(%AA) (1.8)

I now show that any mixed state can be seen as the partial trace of a pure statein a larger Hilbert space. Indeed let %A be a density operator on HA and %A =∑

j λj∑

r |λj,r,A〉〈λj,r,A| be its spectral decomposition. One can define |ψAB〉 =∑j

√λj∑

r |λj,r,A〉 ⊗ |bj,r,B〉 in the tensor product space HA ⊗ HB, with HB beingany dA-dimensional Hilbert space and {|bj,r,B〉} its basis. Then

%A = TrHB(|ψA,B〉〈ψA,B|) (1.9)

This means that mixed states are a consequence of the fact that quantum systemsinteract with one another: if it was possible to truly isolate them, they would remainpure. This is one of the most important obstacles to the realization of a quantumcomputer, in which the environment can inconveniently change the results of calculationsand cause loss of information or errors.

However, since every cloud has a silver lining, this fact can also be exploited to practicallygenerate states of very low purity. If it is possible to produce states that are almost pureand entangled in a bipartite system, one can ignore all measurements in a subsystemand get a mixed state in the other. This technique has been effectively used in theexperiment that will be described in Chapter 4.

It is interesting to observe that after the introduction of the principles of quantummechanics, many generalizations had to be added to better describe the experimentalevidence. Mixed states are one of them, others are for example completely positive maps,which expand the evolution of a system out of the bounds of unitary transformations,or POVMs, which provide a model of measurement that is more realistic than thatof projections. However, in all these cases ideality can be restored by extending thesystem under study. The principles only seem incomplete when we are missing thebigger picture.

Introduction 5

1.3 The Bidimensional Case: The Qubit

In order to simplify the understanding of this formalism and to introduce some impor-tant notations, I will now provide some examples in the easiest possible case: the qubit.This term is vastly used, especially in the field of quantum information, as a synonymfor two-level quantum system, and therefore is applied to many different physical im-plementations, such as spin 1/2 of a particle, currents in superconductors and photonpolarizations. Since the Hilbert space of a qubit is of course bidimensional (d = 2), onecan define a basis with two orthonormal pure states {|0〉, |1〉} and numerically representthem with the unit vectors of their coordinates in C2:

|0〉 =

(10

)|1〉 =

(01

)(1.10)

where an irrelevant phase factor has been neglected. Any complex linear combinationwhich preserves the normalization request is still a pure state:

|ψ〉 =

(αβ

)with |α|2 + |β|2 = 1 (1.11)

From these, it is possible to write the corresponding density matrices:

%0 = |0〉〈0| =(

1 00 0

)%1 = |1〉〈1| =

(0 00 1

)%ψ = |ψ〉〈ψ| =

(|α|2 αβ?

α?β |β|2)

(1.12)

All the aforementioned properties hold true and in these cases Tr(%2) = 1 because theconsidered states are pure.

It is paramount to understand the profound difference between a linear combination ofpure states and one of their density matrices. Let, for example, |+〉 be the pure statewith α = β = 1√

2:

|+〉 =1√2

(11

)→ %+ = |+〉〈+| = 1

2

(1 11 1

)(1.13)

A measurement in the {|0〉, |1〉} basis will result in either of the two outcomes with50% probability, but there exists at least one observable the measurement of which isdeterministically predictable, that is, of course, the one which has |+〉 and its orthogonalstate |−〉 as eigenvectors. Consider instead a combination of %0 and %1 such as

%M =1

2(%0 + %1) =

1

2

(1 00 1

)(1.14)

There is no (non-trivial) measurement for which one outcome has unitary probability,and this state is completely mixed (also said totally unpolarized) and Tr(%2

M ) = 12 . In

order to exemplify the process of purification, I point out that the two-qubit pure state|ψA,B〉 = 1√

2(|0A〉 ⊗ |0B〉+ |1A〉 ⊗ |1B〉) is such that %M = TrHB

(|ψA,B〉〈ψA,B|).

There is another vastly used notation for two-level states. Since all possible densityoperators of qubits are 2× 2 complex hermitian matrices, they can be expressed as real

6 Introduction

linear combinations of the three Pauli matrices σx, σy, σz and the identity 12.

12 ≡ σ0 ≡(

1 00 1

)σx≡ σ1 ≡

(0 11 0

)σy ≡ σ2 ≡

(0 −ii 0

)σz≡ σ3 ≡

(1 00 −1

)(1.15)

The necessity that Tr(%) = 1 fixes the coefficient of 12 at 12 :

% =1

2(12 + ~r · ~σ) (1.16)

where the request for positivity imposes r2 ≤ 1.

This shows that the space of states of two-level systems is in one-to-one correspondencewith a ball in R3 called the Bloch sphere. From (1.16) it follows that Tr(%2) = 1

2(1 + r2),therefore r = 1 for pure states and r = 0 for completely mixed ones. The Bloch spherevisualization is useful not just for purity, but also to find eingenvalues and eigenvectorsof a density operator because

% =

(1 + r

2

)1

2

(12 +

~r

r· ~σ)

+

(1− r

2

)1

2

(12 −

~r

r· ~σ)

(1.17)

where |λ±〉 = 12(12± ~r

r · ~σ) are valid pure states and eigenvectors of %, while 1±r2 are the

corresponding eigenvalues. Given a density operator as a point in the Bloch sphere, itseigenstates are the projections of its direction on the surface of the sphere, whereas inthe case of r = 0 all the states are eigenvectors since % is a scalar matrix.

| +

|

|i

| i

|0

|1

Figure 1.1: A visualization of the Bloch Sphere.

Introduction 7

1.4 Quantum State Tomography

The central objective of this work is the exploration of new alternatives to quantum statetomography, hence I will now briefly explain the general ideas of its standard procedures[18]. Reconstructing a density matrix means finding its d2 − 1 real parameters withappropriate measurements on the system. It is straightforward to imagine that at leastthe same amount of independent pieces of data is needed, and in order to compensate forexperimental errors, measurements must be repeated many times. Since every externalaction changes or even destroys the system, tomography needs a large number of copiesof it, so as to perform one measurement on each.

Provided one can produce these systems, they then need an experimental setup capableof projecting them on d2−1 different states and extract the success probabilities of theseoperations. Moreover, any detector can provide only data that are proportional to suchprobabilities, therefore one needs also a normalization constant, bringing the total countof measurements to d2.

There are many different choices of useful projection states, but a good starting pointis a basis for the complex vector space of d× d matrices, which is, not by accident, d2-dimensional. Let {σk} be such a basis, where the symbol σ is used because in the caseof d = 2 one example is, again, the set of Pauli matrices plus identity {12, σx, σy, σz}.Let Tr(σj σk) = αδj,k, with α a constant. Then for any d× d complex matrix M :

M =

d2∑k=1

α−1σk Tr(σkM) (1.18)

After applying the measurement projectors {Πj | j = 1..d2} to systems all in the statedescribed by density operator %, the observed data are:

nk = N Tr(Πj %) (1.19)

where N is the aforementioned normalization constant. Substituting from (1.18) into(1.19), one obtains:

nj = N Tr

Πj

d2∑k=1

α−1σk Tr(σk%)

= Nα−1d2∑k=1

Tr(σk%) Tr(Πj σk) (1.20)

By defining the d2 × d2 matrix B such that Bj,k = Tr(Πj σk) and by inverting it, onegets:

Tr(σk%) = N−1α

d2∑j=1

(B−1)k,jnj (1.21)

From these values it is possible to reconstruct % via (1.18). In order to find N−1 onesimply has to divide the entire % by its trace so that it becomes equal to 1. It is clearthat projectors {Πj} have to be linearly independent, so that B is invertible: only in thiscase the measurements can carry through the scheme and are said to be tomographicallycomplete.

The example of the qubit can again facilitate comprehension. In 1852, long before theadvent of quantum mechanics, George Stokes proposed a way to experimentally define

8 Introduction

the polarization state of light [19]. A slightly simplified protocol uses the following setof projectors:

Π0 =1

2(|0〉〈0|+ |1〉〈1|) Π1 = |+〉〈+| Π2 = |i〉〈i| Π3 = |0〉〈0| (1.22)

where |i〉 = 1√2(|0〉 + i|1〉) while the other symbols are defined in Section 1.3. Observe

that in this case each of the projectors is present in the spectral decomposition of exactlyone of the basis matrices σk: this is a sufficient but not necessary condition for a set to betomographically complete. Moreover, with this choice the search for N is straightforwardbecause for any valid density operator %

Tr(Π0%) = 1 → n0 = N (1.23)

In this basis α = 1 and

B =

1 0 0 01 1 0 01 0 1 01 0 0 1

→ B−1 =

1 0 0 0−1 1 0 0−1 0 1 0−1 0 0 1

(1.24)

so that the desired coefficients of the Bloch vector are simply

Tr(σk%) =nk − n0

n0(1.25)

In general, B can be much more complicated, especially when the choices of basis andprojectors are limited by experimental feasibility. This means that it is usually neces-sary to perform the entire set of d2 measurements even if one is interested in a singlecoefficient. Besides, for very large d, matrix inversion can become computationally pro-hibitive and the propagation of experimental errors through (1.21) can largely invalidatethe results.

It is common, for example, that the reconstructed matrix is not positive semidefiniteand Tr(%2) > 1. In this case one must use the experimental data in a further numericalminimization step based on χ2 or likelihood functions that force the result to satisfy theproperties of a density operator. This can be done through the definition of a generalvalid density matrix ρ which is a function of d2 real parameters t = t0..td2−1. Thesecan all be contained in a triangular matrix T (t) with real diagonal elements such as (ind = 2):

T (t) =

(t0 t1 + it20 t3

)(1.26)

from which it is possible to obtain any quantum state through expression

ρ(t) ≡ T (t)T †(t)

Tr(T (t)T †(t))(1.27)

Then one can calculate the expectations values Tr(Πj ρ(t)) as functions of the parameters

and minimize χ2 ≡∑

j(nj−N Tr(Πj ρ(t)))Var(N Tr(Πj ρ(t)))

to find topt that better matches the measure-

ments results and provides the best estimate for % = ρ(topt).

Chapter 2

Weak Measurements

In a seminal paper published in 1988 [20], authors Y. Aharonov, D. Albert and L.Vaidman (later AAV) introduced a new way of performing a quantum measurementwhich produced unexpected, strange, yet very useful results. They proposed to weakenthe perturbation on the system caused by its observation so as to prevent or at leastreduce the collapse of its state. The cost of this operation is a greater imprecision in theresults, which has to be compensated for with more repeated acquisitions. Since then,weak measurements have found many theoretical and experimental applications, someof which are clever new schemes of state reconstruction and are hence crucial for thecentral part of this thesis. This chapter is devoted to the detailed explanation of theseideas and to the presentation of the existing direct state measurement protocols that Iwish to evolve.

2.1 Weak Values and the Ancilla Measurement Scheme

AAV started their dissertation by modeling the observation process as an interactionbetween the system under study (or object) and a measuring device, the ancilla (alsopointer, meter). This visualization was introduced first by Von Neumann [2, Ch. 6]and is actually pretty similar to the common intuition of a classical measurement, inwhich the result is read off a tool that is separated from the variable of interest. In thequantum case, though, the important distinction is that between two Hilbert spaces, HS

and HA respectively for object and ancilla, which in practice can also refer to differentproperties of the same physical entity.

Suppose that at the beginning the total system lies in the separable and (for simplicity)pure state:

|ψ〉 = |φS〉 ⊗ |IA〉 (2.1)

and that one wants to measure observable S in the object system. In order to do so,they can couple it to an operator of the ancilla, so that a change in its state reflectsthe action of S on |φS〉. Intuitively, the pointer, just like the needle of a scale, reactsto the observation of S and carries the result of the measurement. Without loss ofgenerality, I assume HA to be continuous, so that an example of coupled operator canbe the canonical momentum PA (conjugated to position XA). This interaction can bemodeled by Hamiltonian

Hint(t) = g(t)S ⊗ PA (2.2)

9

10 Weak Measurements

where g(t) describes its strength and duration. Assuming that the intrinsic Hamiltoniansof each Hilbert space vanish, the system evolves according to unitary operator:

U = e−i~∫Hint(t)dt = e−i

g~ S⊗PA (2.3)

where g ≡∫g(t)dt. Since PA generates translations for states expressed in the xA basis,

the outcome of the measurement of S can be read as a shift in the position of thepointer. For instance let the initial state of the ancilla be a real valued function of xA

with zero mean such as the gaussian IA(xA) ≡ 〈xA|IA〉 = (√

2π∆x)−12 e− x2

4∆2x and let |φS〉

be written in the basis of eigenstates of S as |φS〉 =∑

j cj |sj〉. Then, in the result ofthe evolution |ψ′〉, the pointer position is distributed in a comb of translated gaussianscentered around different gsj : if they do not meaningfully overlap (∆x � g|sj − sk|), ameasurement of XA selects one of them and makes the object system collapse onto oneparticular |sj〉. It is also clear that

〈X(g)A 〉ψ′ = g〈S〉φS (2.4)

in which 〈XA〉IA would appear added to the right-hand side if it was not null as in thepresent example. In this strong case, this model closely represents the axiomatic idea ofquantum measurement.

AAV asked themselves what happens when the strength of the interaction g is smallcompared to the precision of the meter (∆x � g|sj − sk|). In symbols, this means thatthe evolution operator can be truncated to first order in g:

U = e−ig~ S⊗PA ≈ 1− i g

~S ⊗ PA (2.5)

so that the state of the joint system after its action is

|ψ′〉 ≈ |φS〉 ⊗ |IA〉 − ig

~S|φS〉 ⊗ PA|IA〉 (2.6)

which is similar to the initial unperturbed configuration. Of course, now a measurementof XA cannot directly provide a value for sj , because the distribution of results is largeeven if |φS〉 is an eigenstate of S. However, the central moment of this distribution stillcoincides with the mean value of gS as in (2.4) and the precision of its determination canbe improved by repeating the procedure many times on identical systems. I point outthat the literature occasionally uses the term weak average for 〈S〉 when it is measuredin this way with small g.

More interesting features appear when another step is added to the recipe: postselection.This means performing a projective measurement on the object after the interaction andconsidering only those cases in which it collapses on one particular (postselected) state.If this chosen state is |ΦS〉, the pointer lies in

|FΦ,A〉 =〈ΦS |ψ〉〈ΦS |φS〉

≈ |IA〉 − ig

~〈SW 〉Φφ PA|IA〉 (2.7)

where

〈SW 〉Φφ ≡〈ΦS |S|φS〉〈ΦS |φS〉

(2.8)

Weak Measurements 11

is defined as the weak value of observable S. The normalization of |FΦ,A〉 through

division by 〈ΦS |φS〉 also assumes thatg2|〈SW 〉Φφ |

2

4∆2x

� 1.

It is conspicuous that 〈SW 〉Φφ is not necessarily an eigenvalue of S and is not limited bythe bounds of its spectrum. In general it is not even a real number. Nonetheless, it isa measurable quantity and it is easy to show (Appendix A.1) that it can be obtainedfrom the ancilla variables:

<(〈SW 〉Φφ ) ≈ 1

g〈XA〉FΦ,A

=(〈SW 〉Φφ ) ≈ 2∆2x

~g〈PA〉FΦ,A

(2.9)

Assumptions like the reality and null average of the initial meter wavefunction in boththe xA and pA representations have been used here and will be used again in the courseof this work, yet they are inessential and can be relaxed at the cost of complicating theformalism [21, 22].

For completeness and future reference I add that it is easy to generalize the weak valueto the case of a mixed initial state [23]. In the density matrix formalism, if the objectsystem begins in state %S , the weak value is:

〈SW 〉Φ% ≡〈ΦS |S%S |ΦS〉〈ΦS |%S |ΦS〉

(2.10)

whereas the weak average, which is measured as the weak value but without postselec-tion, coincides with the mean value of the observable:

〈SW 〉% ≡ Tr(S%) (2.11)

Even broader generalizations consider non-projective postselection [24], and mixedpointer states [25] but are of minor interest for this thesis.

2.2 The Qubit Pointer

For many applications, the nature of the ancilla is substantially irrelevant. Any prop-erty of the physical system under study or of the environment surrounding it can betreated as a pointer, provided that the coupling with the object observable S can takeplace. For this reason, it is often more convenient to use discrete and finite-dimensionalancillae that simplify the notation and are experimentally accessible. In particular, inthe next sections and in the experimental part of this thesis, qubit pointers will be usedextensively, therefore it is important to render the previous formalism into this newsetting.

The natural meter operators in place of PA and XA are the Pauli matrices σxA, σyA,σzA. Any of them can be used in the coupling but I adopt σyA for consistency with

most of the literature; then complementary observable σxA can take the role of XA. Theinitial state of the pointer is chosen so that it has zero mean and is real in both the σxAand σyA representations, therefore the eigenstate |0A〉 of σzA is a valid candidate.


The evolution operator describing the coupling is:

U = e−iθS⊗σyA (2.12)

which, similarly to the translation of formula (2.3), implements a rotation of the pointerstate around the y axis of the Bloch sphere by an angle that is the result of the measure-ment of S in the object system, multiplied by twice the strength coefficient θ. Repeatingthe same steps of the previous section, the weak value can be measured from the ancillavariables as:

<(〈SW 〉Φφ ) ≈ 1

2θ〈σxA〉FΦ,A

=(〈SW 〉Φφ ) ≈ 1

2θ〈σyA〉FΦ,A

(2.13)

which are valid supposing that θ2|〈SW 〉Φφ |2 � 1.

Summarizing, the reinterpretation of the formalism goes as follows:

PA → σyA

XA → σxA

g → θ

|IA〉 → |0A〉

e−ig~ S⊗PA → e−iθS⊗σyA

(2.14)

The point where this analogy slightly falters is equation (2.4) which becomes

〈σ(θ)xA〉ψ′ =

⟨sin(

2θS)⟩

φS(2.15)

and loses much of its significance outside of the weak regime. The appearance of rotationsand periodic functions such as sin(·) is an indicator of the fundamental limitation of qubitancillae, which lies in the compactness of the Bloch sphere. In the previous formalism ofthe continuous pointer, if the initial and translated gaussians are not separated enoughto be unambiguously distinguishable, one can imagine to increase g to make them fartherapart. In the qubit case this separation is achieved when the initial and rotated stateare orthogonal, but if the measuring device is so imprecise that it still cannot demarcatethem (e.g. a polarizer that transmits some light although it being orthogonally polarizedto its axis), the only solution is enhancing the statistics. In other words, it is uselessto amplify a measurement using an arbitrarily large value of θ, as the pointer statewould simply rotate and come back to where it was before. However, in what followsit is assumed that the ancilla is not affected by this problem or the statistics is richenough to neglect the imprecision in the discrimination of orthogonal states. Then, asdescribed in Chapter 3, strong measurements with qubit pointers regain their usefulness.In particular, if S = ΠS is a projector, the choice of θ = π

2 rotates |0A〉 to its orthogonal

|1A〉 when the projection yields a positive result. The mean value of Π1A carries theinformation of 〈ΠS〉φS so that in place of (2.15) one can write:

〈Π(θ)1A〉ψ′ = sin2(θ)〈ΠS〉φS (2.16)


2.3 Applications of the Weak Value

The possibility of observing a quantum system without perturbing it has elicited exten-sive research around weak values in the last 30 years. Some of it has brought clarity ontomany fundamental questions of the theory, while some has made the weak measurementan important simplifying tool in experiments. This section has the purpose of listingsome of these applications before moving on to those that are essential for this work.

2.3.1 Amplification through Postselection

The weak value was appreciated first as a source of amplification of feeble signals. Thisis mainly due to the fact that if the initial and postselected states almost do not overlap,the real and imaginary parts of 〈SW 〉Φφ can reach great numerical values, making thepointer measurements achievable with equipment of limited sensitivity. From them, itis usually possible to extract interesting physical parameters such as the small couplingconstant g. To clarify, the point is not to use an interaction so strong that even a smallvalue of 〈S〉 can be observed, rather the opposite: the coupling is extremely weak, butthanks to a clever postselection, the product g|〈SW 〉Φφ | becomes measurable. However,this amplification is not arbitrary, as the relations such as (2.9) or (2.13) that govern it

are valid only ifg2|〈SW 〉Φφ |

2

4∆2x

� 1, or θ2|〈SW 〉Φφ |2 � 1 for qubit ancillae: the initial spread

of the meter wavefunction is the fundamental limit of the process.

He-Ne Source

Lens

Polarizer

BirefringentCrystal

Polarizer

CCD Sensor

Lens

Preparation

Weak Measurement

Postselection

Detection

Figure 2.1: Scheme of the first experiment thatexploited the amplifying effects of postselection

[26].

This has not stopped experimentalistsfrom achieving remarkable results, thefirst of which came in 1990 [26] with anoptical setup [27] inspired by the particleanalog originally proposed by AAV [20].Laser light passes through a thin bire-fringent quartz plate which separates twoorthogonal polarizations by a distance athat is much smaller than the beam waist(hence the weakness of the interaction).A subsequent polarizer postselects only afraction of the emitted light and sends itto a CCD. In this case the space of polar-izations is the object system and the roleof S is played by the projector on the ketcorresponding to the displaced beam. Themeasured meter observable is the positionof the light on the sensor, represented byan intensity distribution over its pixels.The mean value of this distribution is pro-

portional to small parameter a, but is greatly amplified by the weak value if the post-selected state is chosen to be nearly orthogonal to the initial one. Therefore it can beobserved and the value of a can be accurately ascertained even if the width of the pixelsis large. In the cited experimental case, the precision of the measurement reached thetens of nms despite the use of 1.4 µm wide pixels.


In 2008 an even more extreme amplification was used to observe the spin Hall effect oflight, a splitting of a beam into two polarized components due to a gradient in refrac-tive index. Using clever choices of initial and final polarization states, and exploitingalso a free evolution of the system before the measurement, experimenters could achievean amplification factor of almost 104 and detected displacements close to the angstrom[28]. Other noteworthy examples reported observations of beam angular deflections inthe hundreds of femtoradians [29] and studies of the Imbert-Fedorov effect with ampli-fications of at least two orders of magnitude [30]. For a more comprehensive list seeReference [31].

Of course these results come at a price. Since their origin is essentially the small overlap〈ΦS |φS〉, only a few events pass the postselection, forcing the acquisition of more datato achieve meaningful statistics. In the above example, this means that although theposition distribution shows an appreciable displacement, its height and integral are lowerunless more measurements are added to compensate.

2.3.2 The Meaning of the Weak Value and the Quantum Three-BoxParadox

Whether 〈SW 〉Φφ is somehow related to the (mean) value of observable S has been thesubject of a lengthy discussion that has shaken the bedrock of the concepts of mea-surement and state [32, 33]. Some light on this question can be shed by the study ofquantum paradoxes such as Hardy’s [34] or the three-box paradox [35]. Focusing forsimplicity on the latter, suppose that a particle can be found in one of three boxes A,B or C and its state is initially described by φ = 1√

3(|A〉+ |B〉+ |C〉). One of the three

projectors ΠA, ΠB, ΠC is measured and then Φ = 1√3(|A〉+ |B〉 − |C〉) is postselected.

Using Bayes’ Theorem it is possible to calculate the probabilities of finding the particlein each box with any of the three intermediate measurements. For instance if the chosenobservable is ΠA, the probability of finding box A populated is

PrA(in A) =|〈Φ|ΠA|φ〉|2

|〈Φ|ΠA|φ〉|2 + |〈Φ|1− ΠA|φ〉|2= 1 (2.17)

Similarly, one can obtain

PrB(in B) = 1 PrC(in C) =1

5(2.18)

which seem absurd because they affirm that both A and B are certainly occupied. How-ever, one has to realize that these values are refer to different projective measurementswhich cannot be performed at the same time.

The problem can be circumvented by the use of AAV’s recipe. For instance, imaginingthat the particle is charged, one can evaluate its presence in any box by sending a probecharge near it, effectively coupling the corresponding projector with the position of theprobe [36]. A measurement of the pointer momentum followed by postselection of Φ willyield the weak values

〈ΠWA 〉Φφ = 1 〈ΠW

B 〉Φφ = 1 〈ΠWC 〉Φφ = −1 (2.19)


The fact that 〈ΠWC 〉Φφ < 0 shows that it is incorrect to interpret these values as indications

of the location of the particle. While a projective measurement of ΠC can only return 0or 1, 〈ΠW

C 〉Φφ can take almost any value, for example if the postselected state is a more

general Φ = sin(θ) 1√2(|A〉+ |B〉) + cos(θ)|C〉 one finds:

〈ΠWC 〉Φφ =

1√2 tan(θ) + 1

(2.20)

which cannot be linked to the number of particles in box C. Then, the solution to theparadox is that although it is true (and experimentally verified [37]) that the weak valuesare those of (2.19), they do not represent actual probabilities of finding a box occupied.

2.3.3 Average Trajectories of Photons in a Double Slit Experiment

During the first lectures of any basic quantum physics course, students learn aboutHeisenberg’s uncertainty principle and its consequences in the double slit experiment.Since |[X, P ]| > 0, it is not possible to observe an interference pattern (connected tomomentum) and simultaneously know the trajectories of the photons (position). Ifone manages to ascertain through which slit a photon has passed, interference and anymomentum information are lost.

But what if this measurement was carried out weakly? In 2011, this idea was put to thetest [38]. Single photons are emitted from a quantum dot device and separated by anin-fiber 50:50 beam splitter, analog to a slit screen. Their polarization (a qubit pointer)is prepared in an initial linear ket and then weakly rotated towards an elliptical stateby a calcite prism depending on the momenta of the photons (coupling). A polarizingbeam displacer (PBD) separates the two circular polarizations along the y direction andsends the results to a cooled CCD. By observing one y-parallel column of pixels at atime (postselection on x) and comparing the intensities of the two ports of the displacer(pointer measurement on σz), it is possible to obtain information on such momenta.Then this analysis can be repeated at different positions on the propagation direction z,thus allowing to map complete spatial trajectories that are correlated to the interferencepattern on the sensor.

QWP

HWP

Mirror

QWP

HWP

Mirror

Polarizer Calcite QWP Lens LensLens PBD CCD

Preparation Postselection and DetectionWeakMeasurement

In-FiberBeam Splitter

Figure 2.2: Scheme of the experimental setup [38].


It is necessary to underline that this result does not violate Heisenberg’s principle, asthese are only average trajectories and not relative to each photon. However, it showsthat weak measurements can provide new insight into a fundamental experiment ofquantum mechanics such as the double slit interferometer.

2.3.4 Monogamy of Bell’s Inequalities Violations

Interactions of strength that is neither maximal nor close to 0 have been recently useful inthe study of entangled systems [39]. Entanglement is possibly “the characteristic trait”of quantum mechanics [40] and its existence has been subject of discussion since theuniversally renowned paper by Einstein, Podolsky and Rosen Can Quantum-MechanicalDescription of Physical Reality Be Considered Complete? [41]. The fact that, in somecases, action on a subsystem can instantaneously trigger change on another was termed“spooky action at a distance”. In 1964, J.S. Bell devised his famous inequality whichcan rule out semiclassical explanations such as local hidden variable models (LHVMs)[42]. If Alice and Bob, the two parts of a bipartite system, can measure two observableseach (A, A′, B, B′), then the CHSH inequality, a variation of Bell’s, reads:

ICHSH = 〈A⊗B〉+ 〈A⊗B′〉+ 〈A′ ⊗B〉 − 〈A′ ⊗B′〉 ≤ 2 (2.21)

A violation of such bound, which has been observed experimentally [43], excludesLHVMs but can be compatible with the predictions of quantum physics: for maximallyentangled states and appropriate choices of observables, ICHSH = 2

√2.

Nonetheless, it is generally accepted that if there are three non-signaling subsystems,it is not possible to simultaneously violate the CHSH inequality in two different pairsof observers Alice-Bob1 and Alice-Bob2 (monogamy) [39, 44]. This is no longer trueif the Bobs can communicate with each other, in particular Bob1 must measure hissubstate before sending it to Bob2 and has to do it weakly, so as not to perturb it much.Supposing that a projector is coupled to a qubit meter, for which the strength of theinteraction between object and ancilla can be described by an angle θ ∈]0, π2 ], it can beshown that with the best choices of initial state and observables

I(1)CHSH = 2

√2 sin2(θ) I

(2)CHSH =

√2(1 + cos(θ)) (2.22)

There is a narrow range around θ = π3 (neither strong, θ = π

2 , nor completely weak,θ → 0) for which both inequalities are violated at the same time. Weak measurementsallow entanglement to be bigamous.

2.4 Direct Weak Measurement of a Pure Quantum State

The most interesting application of weak values for this thesis is of course connected tothe reconstruction of the state of a quantum system. Since this is described by complexnumbers (the coordinates of a unit vector or the elements of a density matrix), it isintuitive that a complex value can be of aid. J. S. Lundeen and his group in Ottawawere the first to notice the connection that I will now report [3].

Suppose that one can prepare a quantum system in a pure state described by unknownunit vector |φS〉 and they want to represent it in the basis {|aj〉 | j = 1..d} of the


d-dimensional Hilbert space HS . The key idea is to weakly measure projector Πaj ≡|aj〉〈aj | on |φS〉 and then postselect on |b0〉 ≡ 1√

d

∑dj=1 |aj〉 in order to obtain

〈ΠWaj 〉

b0φ ≡

〈b0|aj〉〈aj |φS〉〈b0|φS〉

= ν〈aj |φS〉 (2.23)

Provided one can measure 〈ΠWaj 〉

b0φ as explained in Section 2.1, they can find the coordi-

nates 〈aj |φS〉 by scanning over j and keeping |b0〉 fixed. The choice of this postselectionstate is made so that the factor ν does not depend on j and can be eliminated bynormalization of the vector after all its components have been determined.

As if the simplicity of this conclusion was not already stunning, I shall illustrate theexample given by Lundeen et al. and that is often quoted in the literature. Supposeto be interested in the transverse spatial wavefunction of a photon φS(x). In order toperform a weak measurement, it is possible to couple Πx ≡ |x〉〈x| to another variablethat acts as a pointer, for instance the photon polarization. Many copies of the systemare prepared in the initial state

|ψ〉 = |φS〉 ⊗ |0A〉 (2.24)

where |0A〉 labels (for example) the horizontal polarization in the ancilla Hilbert spaceHA and is an eigenstate of σz. For each position xj , the evolution operator that describesthe interaction between object and meter is

U(xj) = e−iθΠxj⊗σy ≈ 1S ⊗ 1A − iθΠxj ⊗ σy (2.25)

in which θ is a small angle. This means that if the measurement of Πxj yields a positiveresult, the polarization is slightly rotated, otherwise it is left as it is.

The connection between these two degrees of freedom can be experimentally achievedusing a small birefringent retarder, a half-wave plate (HWP), that is only hit by thosephotons that are found in the appropriate position xj , and rotates their linear polariza-tion. Of course, a discretization of the theoretically infinite dimensional and continuousHilbert space of transverse modes HS is here carried out by the finite width of the plate,which interacts with a range of positions that is necessarily larger than point-like. More-over, the number of sampled values of x is obviously a finite d, so that the final result isa d-long series of points φS(xj) which is interpolated by the actual wavefunction. Recallthat the objective is a numerical expression of |φS〉 in the basis of positions and not asymbolic formula.

Polarizer

Lens Lens

HWP Sliver

HWPSlit

QWP

PBS

Detectors

Preparation WeakMeasurement

PostselectionReadout

Figure 2.3: Scheme of Lundeen’s experiment [3].


Postselection on complementary variable px, the transverse momentum, can be accom-plished via a Fourier lens and a slit that blocks all photons except for those with px = 0.This means that the postselected state is indeed

|p0〉 ≡1√d

d∑j=1

|xj〉 (2.26)

Finally, the pointer state is described by normalized vector

|Fp0,A〉 =〈p0|U(xj)|ψ〉〈p0|φS〉

≈ |0A〉+ θφS(xj)

φ|1A〉 (2.27)

where the weak value appears asφS(xj)

φand φ ≡

∑dj=1 φS(xj). By choosing the overall

phase of φS so that φ is real and positive, one easily recognizes that

<(φS(xj)) ≈φ

2θ〈σ(j)x 〉Fp0,A

=(φS(xj)) ≈φ

2θ〈σ(j)y 〉Fp0,A

(2.28)

After scanning the values of xj , the factor φ2θ can be eliminated by normalization. In

some experimental realizations, it may be possible to know this proportionality constanta priori or with a limited number of additional measurements, so that one can quicklyfind a single φS(xj) if they are not interested in the entire vector (Section 2.6). This iswhat makes this technique direct and less prone to inaccuracies due to the propagationof experimental errors in the convoluted reconstruction of standard QST.

This method provides an universal recipe that can be applied to any kind of systemprovided one can implement the necessary couplings. Apart from the normalizationproblem, one can determine each coordinate of the state vector with a fixed number ofmeasurements that are always the same regardless of the nature of the system or itsdimension.

The very particular eventuality of φ = 0 can be treated separately using a different |b〉in place of |b0〉. In this case factor 〈b|aj〉 can vary with j but is known and can beeliminated manually from every measurement before normalization.

2.5 Direct Weak Measurement of a General QuantumState

The measurement scheme I have just presented is based on the formal connection (2.23)between the weak value and the unit vector representation, and is therefore applicableonly to pure states. Nonetheless, there exist other protocols, mostly introduced by thesame authors [4], that take care of the mixed case. Here, I am going to present two ofthem: the first provides a way to reconstruct the density operator element by element,whereas the second, aiming at the discrete Dirac distribution, renounces directness butgains in practical simplicity.


2.5.1 Direct Weak Reconstruction of the Density Operator

Suppose to be interested in the matrix elements of unknown density operator % in basis{|aj〉 | j = 1..d} and consider Πajak ≡ ΠakΠb0Πaj , where again |b0〉 ≡ 1√

d

∑dj=1 |aj〉. It

is clear that the desired coefficients are expressed by

〈aj |%|ak〉 = d · Tr(Πajak %) (2.29)

However, Πajak contains the products of non-commuting projectors and is hence nothermitian nor an observable. This means that the right-hand side of (2.29) cannotbe the mean value of a projective measurement as in (1.3), nor it can be obtained inthe ancilla scheme, since regardless of the coupled meter operator, the evolution is notunitary and non-physical.

The solution is offered by coupling each of the three projectors separately to threedifferent ancillae and performing three sequential weak measurements. In the exampleof qubit pointers HA, HB, HC and coupled operators σyA, σyB, σyC , the evolution isgoverned by:

U = e−iθCΠak⊗σyCe−iθBΠb0⊗σyBe−iθAΠaj⊗σyA (2.30)

Reference [45] introduces a clever way to calculate weak averages (or weak values if post-selection occurs) of products, which I extented to the case of qubit pointers (AppendixA.2). It is a general result that

〈ΠWajak〉% = Tr(Πajak %) ≈ 1

θAθBθC〈ΣAΣBΣC〉FABC (2.31)

where Σl ≡σxl+iσyl

2 . In practice, because Σl is not hermitian either, the products

of Pauli operators that appear in 〈ΣAΣBΣC〉FABC have to be measured on differentsubensembles of identically prepared systems. Since these operators act on distinctHilbert spaces, those of the three ancillae, they all commute with one another and theirproducts are observables.

For this protocol to work properly, it is fundamental, according to the authors, thatthe couplings are indeed weak, so that a measurement does not affect the next one.However, as Lundeen himself explains, there is no reason for the last interaction not tobe strong. In other words, one can consider the measurement of Πak as a postselectionon |ak〉 and find the matrix element of interest through the weak value

%jk ≡ 〈aj |%|ak〉 = dTr(Πak %)〈(Πb0Πaj )W 〉ak% (2.32)

This fact makes any practical realization much simpler by reducing to 2 the number ofneeded ancillae and has been used in the first verification of this protocol [46], and inthe experimental part of this work. From (2.31) follows that

<(%jk) ≈d

4θAθBTr(Πak %)

(〈σ(aj)xA σ

(b0)xB 〉Fak,AB − 〈σ

(aj)yA σ

(b0)yB 〉Fak,AB

)=(%jk) ≈

d

4θAθBTr(Πak %)

(〈σ(aj)yA σ

(b0)xB 〉Fak,AB + 〈σ(aj)

xA σ(b0)yB 〉Fak,AB

) (2.33)

where the mean values of the Pauli operators are measured on the final pointer state afterpostselection. Provided some information on the normalization of the results is available


(Section 2.6), this scheme allows to find every density matrix element independently andcan be used with any kind of system of any dimension.

2.5.2 Direct Weak Reconstruction of the Dirac Distribution

Another measurement protocol was also proposed by Lundeen [47] and put to the test[48]. Due to its comparatively easier realization, its verification and extension to thecase of strong couplings is a secondary interest of this thesis.

Let {|aj〉 | j = 0..d− 1} be the basis in which one wants to express % and let {|bl〉 | l =0..d− 1} be its Fourier basis. Consider the non-hermitian operator Djl ≡ ΠblΠaj , then

Tr(Djl%) = 〈aj |%|bl〉〈bl|aj〉 (2.34)

is the discrete Fourier transform of the density operator:

〈aj |%|ak〉 =d−1∑l=0

Tr(Djl%)ei2πl(j−k)

d (2.35)

and coincides with the jl element of the corresponding discrete Dirac distribution. Again,one can measure it by coupling sequentially Πaj and Πbl to two distinct ancillae andfinding the weak average

〈DWjl 〉% = Tr(Djl%) ≈ 1

θAθB〈ΣAΣB〉FAB (2.36)

Since the last measurement does not have to be weak, it can be treated as a postselection:

〈DWjl 〉% = Tr(Πbl %)〈ΠW

aj 〉bl% ≈

1

2θATr(Πbl %)

(〈σ(aj)xA 〉Fbl,A + i〈σ(aj)

yA 〉Fbl,A)

(2.37)

so that only one ancilla is needed.

This technique allows to extract the discrete Dirac distribution element by element,but cannot be considered direct as the previous one if the goal is the density operator.Indeed, equation (2.35) shows that at least O(d) measurements are needed for eachmatrix element, although with the same data one can determine an entire row.

2.6 Normalization of the Results

In the previous section I have shown that it is possible to extract the properties ofthe state under study (e.g. its density matrix elements) directly from measurementson the pointers and known constants such as the dimension of the system and thestrength of the coupling. Nonetheless, there seems to be a problem represented by thepostselection probability (later pps), that appears as Tr(Πak %) in (2.33) and as Tr(Πbl %)in (2.37). The solution is connected to the more fundamental procedure of normalizationof experimental results.

In a general quantum physics experiment, especially in the case of photonics as in Chap-ter 4, the datum usually consists of counts that are proportional to the mean value of a


projector. Let O be an observable and let O =∑

l λlΠl be its spectral decomposition:

the system is projected sequentially using the various Πl and counts

Nl = N〈Πl〉 (2.38)

are recorded, where N is a constant factor representing the total power of the signal. Itis clear that

〈O〉 =∑l

λl〈Πl〉 =∑l

λlNl

N(2.39)

in which NlN takes the role of the probability (in frequentist terms) of projection.

In the specific case of the protocols of Section 2.5, the probabilities that appear in themean values of pointer observables are conditioned on successful postselection, that is

〈O〉Fps =∑l

λlpl|ps =∑l

λlpl∩pspps

(2.40)

where pl∩ps is the joint probability of projection on the l-th outcome of the ancillameasure and postselection on a specific state of the object. Since non-selected countsare discarded, Nl = Npl∩ps, so that∑

l

λlNl

N= 〈O〉Fps · pps (2.41)

which is conveniently the term that appears in (2.33) and (2.37), and makes a separatedetermination of pps unnecessary.

Of course this is only a way of concealing the weak average notation of (2.31) and (2.36),but it is preferable to always consider the last coupling as a postselection because it makesthe experimental realization and the calculations much simpler. To further highlight thisand for ease of writing, in future sections I will often use the symbol

〈O〉ps ≡ 〈Πps ⊗ O〉ψ′ = 〈O〉Fps · pps (2.42)

where |ψ′〉 is the joint state just before postselection and Πps is the postselecting pro-jector. One can imagine that the mean value on the left-hand side is calculated on theunnormalized state after postselection.

However, this has merely shifted the problem from pps to N which is still unknown.The most trivial but lengthy way to find it is to renormalize the density matrix after ithas been determined in its entirety. If this is not possible, then one can focus on thediagonal elements, reducing the number of needed measurements from d2 to d. Finally,some experimental setups may provide an estimate of the signal power, and thus of N ,with a single measure analog to that of Π0 in (1.22).

Unfortunately, this does not apply to the formulas of Section 2.4, since even with theknowledge of N , factor φ remains out of reach. In this case one either has to measurethe complete state vector or must have some a priori information on one of its elementsto calibrate the results.

Chapter 3

Direct State Reconstruction viaStrong Measurements

The fundamental idea on which this thesis stands is that the state reconstruction pro-tocols described in the previous chapter do not need weak couplings and can work in(almost) the same way with strong ones [5, 49]. The ancilla scheme is still necessarybecause it separates the interesting variables from those that are actually observed andprovides an useful formalism with which one can find the needed measurements. Astrong link between object system and pointer device is usually much easier to practi-cally implement and allows to extract more information per measure, making the resultsprecise without the need for a large number of acquisitions. Moreover, the final valuesare bound to the measured quantities without any approximation (such as that of (2.25))and are hence less prone to inherent biases. My goal is to show that the exact versionof Lundeen’s formulas can be used for state reconstruction without critical formal orexperimental complications, and the greater disturbance on the object system causedby strong measurements does not constitute a problem for these applications.

3.1 Evolution of the Basic Protocol to Couplings of Arbi-trary Strength

Despite the fact that the main focus of this work is on evolving the schemes reportedin Section 2.5, it is best to start from the basics and consider the extension of the mostfundamental of Lundeen’s protocols (Section 2.4) to the case of strong measurements.Suppose, again, that one wants to reconstruct the transverse spatial wavefunction of aphoton and does so by coupling projector Πxj to the Pauli operator σy in the Hilbertspace of polarizations that acts as a pointer. The evolution is the same as that of (2.25)but this time θ can take any value in ]0, π

2 ].

U(xj) = e−iθΠxj⊗σy = (1S − Πxj )⊗ 1A + Πxj ⊗ e−iθσy (3.1)

which is now exact. The subsequent steps in the procedure are the same: |p0〉 is posts-elected in the object system and the final normalized pointer state becomes

|Fp0,A〉 =〈p0|U(xj)|ψ〉〈p0|φS〉

=

(1 + (cos(θ)− 1)

φS(xj)

φ

)|0A〉+ sin(θ)

φS(xj)

φ|1A〉 (3.2)

23

24 Direct State Reconstruction via Strong Measurements

Although the coupling is strong, the weak valueφS(xj)

φformally appears again here.

Finally, projective measurements are carried out on the pointer to find that

<(φS(xj)) =φ

2 sin(θ)

(〈σ(aj)x 〉Fp0,A + 2 tan

(θ

2

)〈Π(aj)

1 〉Fp0,A)

=(φS(xj)) =φ

2 sin(θ)〈σ(aj)y 〉Fp0,A

(3.3)

The resemblance to (2.28) is conspicuous. However, one additional pointer measurementis needed, that of Π1, where |1A〉 is defined as the orthogonal state of the initial |0A〉.The entire cost of strong couplings is in this single further step. Again, if one knows the

normalization factor φ2 sin(θ) , they can extract each coordinate of the state vector in a

fixed number of measurements regardless of the dimension of the object Hilbert Space.

3.2 Evolution of the Advanced Protocols

If the most essential model of direct state measurement can be generalized to strongcouplings, it seems natural that the other protocols, those that take care of the mixedcase, should too. In general, this extension is just a matter of applying result (3.1) tothe joint evolution operators (such as that of equation (2.30)) and then finding the mostuseful measurements on the pointers.

3.2.1 The Density Operator Protocol

Let the initial object system be prepared in state %S of unknown matrix elements %jk =〈aj |%S |ak〉, with {|aj〉 | j = 1..d} being an orthornormal basis for the Hilbert space HS .Two ancillae HA and HB are prepared in pure states %A = |0A〉〈0A| and %B = |0B〉〈0B|,so that in the general Htot = HS ⊗HA ⊗HB, the configuration is described by theseparable tensor product

%tot = %S ⊗ %A ⊗ %B (3.4)

After the choice of a particular |aj〉, a coupling is made between Πaj in HS and σyA in

HA: the system then evolves according to UA(aj) = e−iθAΠaj⊗σyA in HS ⊗HA and 1B

in HB. Afterwards, a second coupling involves the other ancilla and Πb0 , with |b0〉 ≡1√d

∑dj=1 |aj〉 as usual. The corresponding evolution operator is UB = e−iθBΠb0⊗σyB in

HS ⊗HB whereas the first pointer remains unchanged. After both interactions, thesystem is in a state described by:

%′tot(aj) = UBUA(aj)%totU†A(aj)U

†B (3.5)

Subsequently, a postselection on |ak〉 occurs on HS , leaving the pointers in a state thatI call %AB, since its derivation is quite lengthy, it is postponed to Appendix A.3.

At this point, %jk can be extracted through different measurement strategies on theancillae. Some of them only work for particular values of d or of θA,B, but othersare general. It is certainly possible to extend Lundeen’s formulas of (2.33) (which arestill valid in the limit of θA,B → 0) to the case of arbitrary strength by adding some

Direct State Reconstruction via Strong Measurements 25

measurements on %AB:

<(%jk) =d

4 sin θA sin θB

(〈σ(aj)xA σ

(b0)xB 〉

ak − 〈σ(aj)yA σ

(b0)yB 〉

ak

+ 2 tanθB2〈σ(aj)xA Π

(b0)1B 〉

ak + 2 tanθA2〈Π(aj)

1A σ(b0)xB 〉

ak + 4 tanθA2

tanθB2〈Π(aj)

1A Π(b0)1B 〉

ak

)=(%jk) =

d

4 sin θA sin θB

(〈σ(aj)yA σ

(b0)xB 〉

ak + 〈σ(aj)xA σ

(b0)yB 〉

ak + 2 tanθB2〈σ(aj)yA Π

(b0)1B 〉

ak

)(3.6)

Yet much easier results exist:

%jj =d2

sin2 θA sin2 θB〈Π(aj)

1A Π(b0)1B 〉

ak ∀k

<(%jk) = − d

2 sin θA sin θB〈σ(aj)yA σ

(b0)yB 〉

ak j 6= k

=(%jk) =d

2 sin θA sin θB〈σ(aj)xA σ

(b0)yB 〉

ak j 6= k

(3.7)

The abundance of superscripts (ak) means that postselection probability Tr(Πak %) is leftimplicit. As previously explained in Section 2.6, this factor is not problematic as it canbe absorbed in the measurement of the pointer observables if joint probabilities (outcomeand postselection) are used in place of conditioned ones or if the density operator canbe normalized after its determination.

It is clear that, again, the cost of strong couplings is represented by measurements of Π1

on the two ancillae. It is also noticeable that, as usual, σy appears in the determinationof the imaginary parts of %jk, while its complementary observable σx is necessary for thereal parts.

3.2.2 The Dirac Distribution Protocol

Albeit less direct, the second of Lundeen’s schemes is still worthy of study because ofits practical feasibility, improved by the fact that it needs only one ancilla. Just as inthe previous case, a coupling has to be made between the pointer, in particular its σyAoperator, and projector Πaj in the object system:

%tot = %S ⊗ %A → %′tot = e−iθAΠaj⊗σyA %toteiθAΠaj⊗σyA (3.8)

Then, state |bl〉 ≡ 1√d

∑d−1j=0 |aj〉e

i 2πjld is postselected, and the pointers associated to

systems that pass are measured. I denote:

ρA1(jl) ≡ 1

2 sin(θA)

(〈σ(aj)xA 〉

bl + i〈σ(aj)yA 〉

bl)

ρA2(jl) ≡ 1

sin(θA)〈Π(aj)

1A 〉bl

(3.9)

where ρA2(jl) is independent of l and superscripts (bl) imply that mean values are to bemultiplied by Tr(Πbl %S). If the interest is actually in the Dirac distribution, its elementscan be found as:

Djl = ρA1(jl) + tan

(θA2

)ρA2(jl) (3.10)


However, after looping over all l and choosing one k ∈ 0..d− 1

%jk = d tan

(θA2

)δjkρA2(jl) +

∑l

e2πil(j−k)

d ρA1(jl) (3.11)

This procedure can be repeated with different j or k in order to reconstruct the entiredensity operator.

3.2.3 A Summary of the two Protocols

From now on these two protocols will be termed DRDO (after direct reconstructionof the density operator) and DRDD (direct reconstruction of the Dirac distribution).The second will be considered an indirect way to obtain the density matrix, with theadvantage of using one less ancilla compared to the first. Here is a brief summary oftheir steps.

DRDO DRDD

1. Preparation of the pointer in state%A ⊗ %B.

2. Choice of j, coupling between Πaj

and σyA and subsequent evolution.

3. Coupling between Πb0 and σyB andsubsequent evolution.

4. Choice of k and postselection of state|ak〉.

5. Extraction of %jk as in (3.7) throughmeasurements of 〈σx〉ak , 〈σy〉ak and〈Π1〉ak on the ancillae.

6. Repetition of steps 2-5 with differentj, k if more than one matrix elementis needed.

1. Preparation of the pointer in state%A.

2. Choice of j, coupling between Πaj

and σyA and subsequent evolution.

3. Choice of l and postselection of state|bl〉 of the Fourier basis.

4. Extraction of ρA1(jl) and ρA2(jl) asin (3.9) through measurements of〈σx〉bl , 〈σy〉bl and 〈Π1〉bl on the an-cilla.

5. Repetition of steps 3-4 with all thedifferent l.

6. Choice of k and extraction of %jk asin (3.11) from the results of step 5.

7. Repetition of only step 6 with differ-ent k if more than one matrix ele-ment is needed in the j-th row.

8. Repetition of steps 2-6 with differ-ent j, k if other matrix elements areneeded.

Table 3.1: List of steps of the DRDO and DRDD protocols.

The step of normalization of the results is implicit in the determination of pointer meanvalues.


3.3 Comparison with Weak Reconstruction Protocols

As previously stated, strong measurements are usually easier to practically implementbecause they are in essence equivalent to standard projections. In the next chapter Iwill show that even if one wants to visualize the role of the pointer with the use ofa quantum system that plays it, reducing the strength of the coupling is not alwayssimple. However, this section is devoted to explaining the intrinsic advantages offeredby a greater strength in the measurement schemes discussed so far, in terms of accuracy(lack of biases) and precision (small statistical errors). As usual, the focus is mainly onthe DRDO protocol.

3.3.1 The Inherent Bias of Weak Measurements

Lundeen’s direct reconstruction method is summarized by equation (2.33) that is herereported:

<(%jk) ≈d

4 sin(θA) sin(θB)

(〈σ(aj)xA σ

(b0)xB 〉

ak − 〈σ(aj)yA σ

(b0)yB 〉

ak)

=(%jk) ≈d

4 sin(θA) sin(θB)

(〈σ(aj)yA σ

(b0)xB 〉

ak + 〈σ(aj)xA σ

(b0)yB 〉

ak) (3.12)

This is still a first order approximation in parameter sin(θA) sin(θB) and has to becorrected as in equation (3.6) or replaced as in (3.7) when the coupling strength cannotbe considered close to 0. It is important to study the importance of this bias for growingstrength to understand the errors of a realization of this method. One way to do so isto compute the trace distance between the measured result and the actual state. Letthese be called respectively %W and %, then this distance is defined as:

t(%W , %) ≡ 1

2Tr√

(%W − %)†(%W − %) (3.13)

It is possible to find (Appendix A.3) the elements of %W − % as

%Wjk − %jk = (cos(θA)− 1)δjk%jk +cos(θB)− 1

d

∑l

%jl +(cos(θA)− 1)(cos(θB)− 1)

d%jj

(3.14)and notice that they vanish for small θA,B, thus validating Lundeen’s results. Nonethe-less, the presence of term

∑l %jl makes %W manifestly non-hermitian, against the very

definition of density operator. In order to improve the result, one can take the hermitianpart of %W and normalize its trace to 1:

%WHN ≡%W+%W†

2

Tr(%W+%W†

2

) (3.15)

There is still no guarantee that this matrix is positive semidefinite, but since it hasunit trace, the comparison offered by t(%WHN , %) is more meaningful, moreover it isrealistic to assume that an experimenter who measures the entire %W would convert itto %WHN also to make the procedure more robust to statistical and systematic errors ofthe implementation.


The analytic calculation of t(%WHN , %) is not trivial, therefore I built a simulation (de-tailed in Appendix A.5.1) that randomly generates a valid state % and computes %WHN

and t(%WHN , %) for different values of θ ≡ θA = θB.

0.0 0.2 0.4 0.6 0.8 1.0 (units of /2 rad)

0

2

4

6

8

10

12

14

Trac

e di

stan

ce t

(a) Average trace distance. Verticalbars stretch to the minimum and max-

imum values.

0.0 0.2 0.4 0.6 0.8 1.0 (units of /2 rad)

0.0

0.2

0.4

0.6

0.8

1.0

Prob

abilit

y th

at t

>0.

1(b) Probability that t > 0.1.

Figure 3.1: Results of the simulation for 104 different bidimensional states, eachmeasured with 200 different values of θ ∈]0, π2 [.

As expected the trace distance grows with the strength. Figure 3.1b shows that thismethod is approximately sure to provide accurate results (i.e. t(%WHN , %) ≤ 0.1) ifθ . 0.15π, which is an experimentally viable value. However it is also clear that themethod is off the mark for θ close to π

2 which precludes its use with more convenientstrong measurements. In particular, values t(%WHN , %) > 1 in Figure 3.1a mean thatthe produced matrix is non-physical and does not contain any information from which% can be recovered.

3.3.2 The Role of Experimental Errors

When I introduced weak measurements in Section 2.1, I underlined that the distributionof results is large compared to its mean value and that many repetitions are neededto achieve meaningful precision. A similar problem affects the DRDO protocol if thestrength of the coupling between object and ancilla is kept small. Equation (3.7) showsthat for non-diagonal elements, the measured quantities are mean values in the form

〈σ(aj)µ σ(b0)

ν 〉ak =∑

a,b=±1

ab〈Π(aj)µa Π

(b0)νb 〉

ak (3.16)

where µ, ν ∈ {x, y}, Πµa and Πνb are the elementary projectors that appear in the spec-tral decomposition of σµ and σν and the mean values are calculated on the unnormalized(hence the superscript) pointer state after postselection 〈ak|%′tot(aj)|ak〉.

As mentioned before, an experiment only measures counts Nµa,νb,jk = N · 〈Π(aj)µa Π

(b0)νb 〉

ak ,with N representing the total power of the signal. These results are affected by Poisso-nian statistical error

δNµa,νb,jk =

√N〈Π(aj)

µa Π(b0)νb 〉ak (3.17)


so that, using equation 3.16, error δµν,jk on 〈σµσν〉′j,k is

δµν,jk =1

N

√ ∑a,b=±1

(δNµa,νb,jk)2 =

√Tr(〈ak|%′tot(aj)|ak〉)

N(3.18)

which does not depend on µ, ν but can change with j, k. Here I am not considering thestandard deviation of N , which depends on how it is measured. Although it would bewrong to neglect it in practice, its contribution is usually similar to that of the othercounts, so that the order of magnitude of the global error is the same with or withoutit. See Appendix A.3 for a quick proof of the last equation and the detailed expressionof the trace.

It is interesting to observe that the value of δµν,jk does not exhibit any strong dependenceon θA,B (Figure 3.2a) and it can even be shown that the propagation of this error on allthe matrix elements %jk is √∑

j,k

δ2µν,jk =

√d

N(3.19)

regardless of the strength of the coupling or the object state. This seems to indicate thatstronger measurements do not improve precision. Yet, this picture changes when oneconsiders that the relevant quantities are not 〈σµσν〉′j,k but rather the density matrix

elements, for which the error grows by factor d2 sin(θA) sin(θB) and is clearly bigger in the

weak case. The expression for the relative errors is:

δ(<(%jk))

|<(%jk)|=

d

2 sin(θA) sin(θB)


N

1

|<(%jk)|δ(=(%jk))

|=(%jk)|=

d

2 sin(θA) sin(θB)


N

1

|=(%jk)|

(3.20)

In Lundeen’s method, two mean values of the form 〈σµσν〉′j,k are averaged, thus making

these results smaller by coefficient 1√2.

The presence of factor 1sin(θA) sin(θB) reflects the fact that the density matrix elements

are calculated from linear combinations of counts that are small compared to the countsthemselves. Inverting equation (3.7) (again for non-diagonal elements) shows that forinstance ∣∣∣∣∣∑

ab

abNya,yb,jk

∣∣∣∣∣ =2 sin(θA) sin(θB)

dN |<(%jk)| (3.21)

In the weak limit, this quantity is much smaller than N and therefore vulnerable tosystematic experimental errors, as even a slight proportional bias on one of the countsNµa,νb,jk can radically change its value.

Different considerations apply to the case of diagonal elements, for which the dependenceof the standard deviation on the strength is more evident, as only projector Π1AΠ1B hasto be measured and

δ(%jj)

%jj=

d

sin(θA) sin(θB)

1√N%jj

(3.22)

Here even the counts are small in the weak limit, thus making their relative errors morerelevant.


0.0 0.2 0.4 0.6 0.8 1.0 (units of /2 rad)

0.685

0.690

0.695

0.700

0.705

,01

N

(a) Standard deviation of a pointerobservable needed for element 01.

0.0 0.2 0.4 0.6 0.8 1.0 (units of /2 rad)

0

50

100

150

200

250

300

350

400

((

01))

|(

01)|

N

DRDOLundeen

(b) Relative error on the real part ofdensity matrix element 01.

Figure 3.2: Results of the simulation for 104 different bidimensional states, eachmeasured with 200 different values of θ ∈]0, π2 [. The choice of matrix element %01 is notparticularly relevant. The weak method shows better precision for equal strength onlybecause it averages two pointer measurements. The high values on the vertical axis of3.2b are due the fact that the contribution of N−

12 is neglected. Error bars reflect the

standard deviation associated to the mean over the 104 samples.

The use of strong measurements, allowed by exact formulas such as (3.7), makes theDRDO protocol reach levels of precision (Figure 3.2b) that are unthinkable for weakcouplings without building up the statistics. The only price to pay is, again, the need toimplement projector Π1, which can cause an additional complication in the apparatus.Moreover, an accurate estimate of the coupling strength is needed in all the evolved pro-posals, while Lundeen’s methods can ignore it if the entire density operator is measured,since the value of θ appears only as a multiplicative constant which can be eliminatedby normalization. Nonetheless, the fact that the global number of needed pointer mea-surements is lower (one observable for each diagonal element, two for each non-diagonalelement, against twice this amount in equation (2.33)), should make the experimentalexecution faster, an important advantage for increasing the strength.

3.4 Comparison with QST

Compared to the standard techniques of QST, the protocols described so far have theaforementioned advantage of directness: if a normalization is available, it is possible tofind one matrix element with a very limited number of observations. For instance, theDRDO protocol can extract the real or imaginary part of a non-diagonal element of thedensity operator with only 4 different acquisitions, regardless of the dimension d of theobject system.

Even when the entire reconstruction is needed, as is generally the case, QST can beinconvenient because it needs to implement a set of d2 linearly independent projections.Experimentally, this is usually achieved with only one projecting device and d2 unitarytransformations that change the input state. Indeed if only Π0 is feasible in the set of{Πj | j = 0..d2 − 1} necessary projectors, one has to find {Uj | j = 0..d2 − 1} such that

Π0 = UjΠjU†j . In this way

Tr(Π0Uj %U†j ) = Tr(Πj %) (3.23)


This is not an easy task. One of the simplest possible cases is that of the polarization ofa photon, which will also be treated in detail in the next chapter. To find it, one usuallyprojects on the horizontal, diagonal and left-circular polarizations, and then calculatesthe normalization factor using a fourth direction, such as the vertical one, or with a rawmeasure of intensity (such as that of Π0 in (1.22)). If only one polarizer, for example on|H〉, is available, then a HWP is necessary to project on |D〉 and a quarter-wave plate(QWP) for |L〉. Every new linearly independent projection complicates the apparatus.The QST of this system is a standard procedure that can be carried out with modernequipment in a few seconds, but this may no longer be true if d becomes considerablylarger than 2.

In the case of direct reconstruction protocols, this problem is reduced. The experimentalsetup must only extract from the system the d components corresponding to the statesin the measurement basis, by implementing {Πaj | j = 1..d} for the first coupling andpostselection, plus one additional projection on |b0〉 for the second coupling. If the basis{|aj〉} is chosen cleverly, it is possible to find one unitary transformation U such thatΠ0 = U jΠj(U

†)j . Consequently, regardless of d, the number of unrelated fundamentalunitaries (that is, not counting powers of the same) is only 2, one is U and the other isneeded for Πb0 .

The cost of this is represented by the two ancillae, which can sometimes complicate thesystem, thus making it more vulnerable to experimental errors. Moreover, they surelyincrease the number of required observations by a factor of 8 for non-diagonal elements(4 for the real part plus 4 for the imaginary part). However, only three different bases ofthe Hilbert space of each pointer are necessary, so that the number of transformationsis again independent of d.

QST DRDO

Number of measurements for diag-onal elements

d d

Number of measurements for non-diagonal elements

d2 − d 8(d2 − d)

Total number of measurements d2 8d2 − 7d

Different unitary transformations d2 d+ 1 on the system5 on each pointer

Unrelated unitary transformations d2 2 on the system3 on each pointer

Table 3.2: Number of different measurements that are needed to reconstruct a den-sity operator in a d-dimensional Hilbert space. The DRDO protocol is assumed to

implement two bidimensional ancillae.

In conclusion, when d is large, the DRDO protocol allows experimenters to reconstructthe density operator with fewer different manipulations, so that the number and cost ofcomponents is reduced.


3.5 Measuring Products of Non-Commuting Observableswith Strong Couplings

The extension of weak measurement methods to strong couplings is not necessarilylimited to the field of state reconstruction. In this section I will briefly deviate from themain course of this thesis and show how the ancilla scheme can provide insight into themeasurement of products of non-commuting observables.

Suppose to be interested in the mean value of the product of N observables on state %S ,that is: ⟨

N∏j=1

Oj

⟩%S

= Tr

N∏j=1

Oj %S

(3.24)

If these observables do not commute with one another, their product is not hermitian.However, this mean value can still be measured by sequentially coupling the Ojs (inreverse order) to different ancillae, as suggested by the DRDO protocol of Subsection2.5.1. The main result exploiting weak measurement has already been mentioned informula (2.31) and is detailed in Appendix A.2:⟨

N∏j=1

Oj

⟩%S

≈ 1∏j θj

⟨∏j

Σj

⟩FN

(3.25)

where |FN 〉 is the joint state of all N pointers after all the evolutions and Σj is the

operatorσx+iσy

2 on the j-th qubit ancilla Hilbert space.

As in the previous sections, this relation can easily be made exact and extended toarbitrary coupling strength θ when all the observables are projectors. It is sufficient tomeasure on each pointer the operator

E ≡ σx + iσy2

+ tan

(θ

2

)Π1 (3.26)

then ⟨N∏j=1

Πj

⟩%S

=1∏

j sin(θj)

⟨∏j

Ej

⟩FN

(3.27)

Interestingly, it is also possible to find the mean value of each projector, simply bymeasuring E on the associated pointer and

E′ ≡ Π0 + tan

(θ

2

)σx + tan2

(θ

2

)Π1 (3.28)

on all the others. Indeed

〈Πk〉%S =1∏

j 6=k sin(θj)

⟨Ek∏j 6=k

E′j

⟩FN

(3.29)

The study of non-commuting observables has always been a profound problem of quan-tum physics due to Heisemberg’s uncertainty principle. Of course, this method does notviolate the basic axioms of the theory, because the results are obtained statistically and


not from a single object. However, this is different from simply measuring each projectoron disjoint subensembles of identical systems, because each acquisition contains infor-mation on all of them. For instance, the measurement of Π1 on the k-th ancilla appearsin both Ek and E′k, therefore it contributes to 〈Πk〉%S and to all 〈Πj 6=k〉%S . Similar pro-cedures have been proposed and tested in the weak regime [50, 51], but the use of exactformulas such as these might provide more accurate results and reduce the statisticalerrors.

The derivation of these expressions is reported in Appendix A.4. As in the weak case,the fact that E and E′ are not hermitian is not problematic because it is possible tomeasure the various pointer observables (σx, σy, Π0 and Π1) one by one and then sumthe results.

I conclude adding that if the focus is on products of observables that are not projectors,one has to spectrally decompose them and apply this procedure to all the differentcombinations. Of course this means that the number of measurements grows with dN ,but this technique can still be convenient in some experimental cases. For instanceif multiple apparati that can measure the various Ojs are available, it may be betterto exploit them than to handle the possibly much different product operator (or itshermitian and antihermitian parts if it is not an observable).

Chapter 4

The Experiment

The bulk of this work is the experimental realization of an apparatus that can reproducethe steps of the protocols reported in Section 3.2. Designed and built by Luca Calderaroand myself at the CNR-IFN Luxor laboratory in Padova, the setup that will be describedin this chapter aims to reconstruct the polarization state of a photonic system usingoptical devices. In order to implement the ancilla scheme we used PBDs to couple theobject to another qubit, the path taken by each photon in a Mach-Zehnder interferometerthat acts as our pointer. This choice of degrees of freedom is dictated by the availabilityof components and by the expertise of our lab and research group. Of course, the factthat the object Hilbert space is only bidimensional puts our measurement schemes ata disadvantage with respect to standard QST, which can reconstruct any polarizationdensity operator with only d2 = 4 data points. However, this experiment is to beinterpreted as a proof of concept, the goal of which is to show that direct reconstructioncan be performed with strong measurements with improved results compared to thoseobtained in the weak case. For this purpose, our setup can change the strength of theinteraction with the ancilla, making it possible to test our and Lundeen’s protocols inboth regimes.

Upstream of the measurement system proper we used a custom built source that ex-ploits spontaneous parametric down-conversion (SPDC) to produce pairs of polarization-entangled photons of wavelength λ ≈ 809 nm. Compared to a continuous wave laser,this source has the disadvantage of lowering the power of the signal from some mW sto the thousands of photons per second (power ∼ 10−16 W ), making the use of singlephoton avalanche diodes (SPADs) necessary. These solid state detectors employ the im-pact ionization mechanism of reversely biased p-n junctions to induce avalanche currentsevery time they are hit by one photon. With the aid of two of these devices (ExcelitasSPCM-AQRH-14-FC) and a fast time-tagger (qutools quTAU), it is possible to recognizethe pairs from the difference in detection time: we used a 1 ns-wide window to definecoincident counts that can be attributed to the entangled photons. Considering onlythese events, we could operate in true quantum conditions and we could easily producemixed states by sending one member of each pair to the measurement setup and usingthe other only as a herald, so that its contribution to the joint state is effectively tracedout.

After the detailed description of this source and of the principles on which it is based, thischapter will focus on the process of planning, building and calibrating the measurementapparatus.

35

36 The Experiment

4.1 The Source of Entangled Photons

Nowadays, entanglement is a fundamental piece of many quantum physics experimentsbecause it is strikingly alien to the classical framework and permits to investigate themost mysterious aspects of the theory. It can exist between degrees of freedom of thesame entity or in two separated ones, the latter case being the most interesting forquantum communication because it allows non-classical correlations to move betweenparties that are far apart.

Modern techniques can generate entangled states with atoms [52], ions [53] or supercon-ductors [54], but the simplest and most widely used case is that of photons, in whichcorrelated pairs are created through SPDC. However, correlation is not the only ingre-dient: the other is the superposition of (at least) two generation processes, so that itis not possible to determine from which one the photons have originated. I will nowexplain how our source achieves these two goals.

4.1.1 Correlation: The SPDC process

Spontaneous Parametric Down Conversion originates from the interaction between lightand matter and exploits the non-linear optical properties of certain materials. Let ~Eand ~B label the classical components of an EM field. Their effects on a medium aredescribed by Maxwell’s equations:

~∇ · ~D(~r, t) = ρ(~r, t)

~∇ · ~B(~r, t) = 0

~∇× ~E(~r, t) +∂ ~B(~r, t)

∂t= 0

~∇× ~H(~r, t)− ∂ ~D(~r, t)

∂t= ~J(~r, t)

(4.1)

where ~D = ε0~E+ ~P and ~H = 1

µ0

~B− ~M are respectively the displacement and magnetizingfields. Since most optical components are uncharged, non-conducting and non-magnetic,the density of free charges ρ, that of free currents ~J and the magnetization vector ~Mare all null, so that the interaction is entirely captured by the polarization vector ~P .

If χ is the non-linear electric susceptibility of the medium, then

Pi = ε0

∑j

χ(1)ij Ej +

∑jk

χ(2)ijkEjEk +

∑jkl

χ(3)ijklEjEkEl + . . .

(4.2)

in which terms of order higher than 1 can give rise to EM fields at different frequencies.It is possible to write the Hamiltonian density as

H =1

2

(~E · ~D + ~B · ~H

)=

1

2ε0| ~E|2 +

1

2µ0| ~B|2 +

ε0

2

∑ij

χ(1)ij EiEj +

ε0

2

∑ijk

χ(2)ijkEiEjEk + . . .

= H0 +HI

(4.3)

in which HI describes the interaction between fields and matter.

The Experiment 37

Without delving too much into the details of quantum field theory, suffice it to say thatfields can be transformed into operators as prescribed by second quantization so thatthe interaction Hamiltonian density becomes

HI ∝∑ij

χ(1)ij

(E

(+)i + E

(−)i

)(E

(+)j + E

(i)j

)+∑ijk

χ(2)ijk

(E

(+)i + E

(−)i

)(E

(+)j + E

(i)j

)(E

(+)k + E

(i)k

)+ . . .

(4.4)

where E(+) and E(−) can respectively annihilate and create a field [55]. One particularterm of the Hamiltonian is responsible for the pair generation:

HSPDC =

∫d3r

∑ipjski

χ(2)ipjski

E(+)ip

E(−)js

E(−)ki

(4.5)

in which symbol p stands for the annihilated pump photon, whereas i and s label thecreated idler and signal photons.

In order for the flow of energy to steadily move from the pump to the produced photonsduring the propagation in the medium, the so-called phase matching conditions have tobe valid:

ωp = ωs + ωi

~kp = ~ks + ~ki(4.6)

If the second is not verified, which can happen because of dispersion, the SPDC processis canceled out by its opposite an the net efficiency stays low. There are various ways toachieve phase matching, some of which are connected to the polarization of the involvedlight:

• Type I birefringent phase matching. A birefringent crystal is used, the pump seesthe extraordinary refractive index while idler and signal see the ordinary one (orviceversa). This means that the produced photons share the same polarizationwhich is orthogonal to that of the pump. One way of verifying phase matching is:

ωs = ωi and no(ωs) + no(ωi) = 2no(ωs) = 2ne(ωp) (4.7)

Photons are emitted symmetrically in a cone the axis of which coincides with thepropagation direction of the pump.

• Type II birefringent phase matching. A birefringent crystal is used, pump andsignal share the same polarization, orthogonal to that of the idler. One way ofverifying phase matching is:

ωs = ωi and ne(ωs) + no(ωi) = 2ne(ωp) (4.8)

Photons are emitted in two different cones.

• Quasi phase matching. The crystal is engineered so that the sign of χ(2) changesperiodically and the production contributions always sum constructively. This canbe achieved also without birefringence and with only one polarization involved (aconfiguration known as type 0 phase matching).

In all cases the polarization states are correlated with one another.

38 The Experiment

4.1.2 Superposition: The Sagnac Interferometer

The polarization and direction of the emitted photons suggest different strategies toachieve the superposition necessary for reliable entanglement generation. For instance,two type I crystals can be paired with their optical axes orthogonal, so that the incomingpump light can interact with the same probability with each one [56]. A single crystalis sufficient for type II, provided photons are collected from the intersection of the twoexit cones [57].

|H⟩+|V⟩

|H⟩

|V⟩

|HH⟩+|VV⟩

Two crystals

with perpendicular

optical axes

Light polarized

at 45° compared

to the optical axes

Entangled state

(a) Type I.

|H⟩|HV⟩+|VH⟩

Only one crystal

Light polarized

at 0° compared

to the optical axis

Entangled state

(b) Type II.

Figure 4.1: Visualization of the pair production process for entanglement generation.

Our source, developed by Dr. Matteo Schiavon as part of his PhD project [58], employsa KTiOPO4 crystal (RAICOL Crystals PPKTP) to achieve quasi phase matching, butthe produced polarization states have the properties of type II SPDC (idler and signalare orthogonally polarized). However, superposition is realized in a way different fromany of those of Figure 4.1.

The pump is provided by a CW laser diode (Ondax LM-405) of nominal wavelength405 nm which enters a Sagnac interferometer through a polarizing beam splitter (PBS).In this configuration the two arms share the same physical path and OPL, hence cannotintroduce a phase difference. The polarization of the clockwise arm is soon switchedfrom |V 〉 to |H〉 by a HWP, so that both beams are horizontally polarized when theyhit the crystal. Some of the photons from each arm trigger the SPDC process and areconverted into pairs that are orthogonally polarized and have approximate wavelength809 nm. Those of the counterclockwise path encounter the HWP (which is designedto work at both wavelengths) and exchange their polarizations. At the PBS, the twoidler photons proceed towards the A exit and the two signals towards B, where theyare collected into two single mode fibers and sent to the rest of the experiment. Thesuperposition request is fulfilled by the fact that it is impossible to ascertain from whicharm each of the signal (or idler) photons have come from. The pump light exiting theinterferometer from the entrance port is deflected by dichroic mirrors and filtered out.

This source can generate maximally entangled states such as the singlet

|Ψ−〉 =|HAVB〉 − |VAHB〉√

2(4.9)

Indeed, just before the fibers, the state is described by

|ψpre〉 = a|HAVB〉 − eiφb|VAHB〉 (4.10)

The Experiment 39

Laser Pump

SM Fiber

SM Fiber

SM Fiber

Dichroic Mirror

Rotating QWP and HWP

Lens

Long-PassFilter

Long-PassFilter

PBS

Non-LinearCrystal

A Exit

B Exit

HWP

Figure 4.2: Scheme of the source. For reasons of mobility, the laser pump is separatedfrom the rest of the setup and connected to it via a single mode (SM) fiber. Originally,a polarization maintaining fiber was used in its place, which allowed to keep the linearpolarization of the pump. Unfortunately, it had to be replaced due to damage and a

QWP was added to better tune the phase between the |H〉 and |V 〉 components.

where a, b, φ are real parameters that depend on the polarization of the pump lightentering the interferometer and a2 + b2 = 1. By rotating the HWP before the PBS,it is always possible to tune a and b so that a = b = 1√

2, then it is only necessary to

compensate for φ and for the effects of the two exit fibers. Supposing that these applyarbitrary unitary transformations UA and UB to the states of the photons that passthrough them, the polarization after the fiber is

|ψpost〉 = (UA ⊗ UB)|HAVB〉 − eiφ|VAHB〉√

2(4.11)

If further transformation U = UA

(1 00 eiφ

)U−1B is applied to the light exiting the B

fiber (for example with additional waveplates), the final state is

|ψfinal〉 = (12 ⊗ U)|ψpost〉 = eiφUA|HA〉 ⊗ UA|VB〉 − UA|VA〉 ⊗ UA|HB〉√

2

= (UA ⊗ UA)|Ψ−〉 = |Ψ−〉(4.12)

where the last equation is justified by the fact that (UA ⊗ UB)|Ψ−〉 = |Ψ−〉 as long asUB = UA.

40 The Experiment

However, for our experiment, the production of such a precise state is not necessary, itis sufficient to obtain states in the form of equation (4.10) with some control over a andb offered by the HWP.

4.1.3 Performance and Polarization Stability

The number of produced pairs depends on many different factors, the most importantof which is of course the power of the pump laser. Due to the small χ(2) coefficient, onlya tiny fraction of the photons that hit the crystal is converted. Of the results, manyare lost in the fiber coupling and some are not detected because of the imperfect SPADefficiency (nominally ηSPAD ∼ 45%).

We analyzed the production for different power values by counting photons just afterthe two exit fibers of the source. The ratio between coincident and total counts isηcoinc ≈ 7%, which suggests that the efficiency of the fiber coupling is around ηcoupling =ηcoincηSPAD

≈ 16%. Such a small value is mostly due to the fact that the signal and idlerbeams are not collimated because of the lens that focuses the pump light at the center ofthe crystal. This compromise was made as it greatly improves the production efficiencyof the material itself.

Driving Current(mA)

Pump power onthe crystal (µW )

Photon kilo-counts/s

Coincident Pho-ton kilo-counts/s

20 10.76± 0.01 2.8± 0.1 0.19± 0.01

25 343± 1 116± 1 10.3± 0.1

30 2302± 2 598± 1 44.3± 0.2

35 3331± 1 961± 5 67.2± 0.4

40 6840± 1 1580± 7 106.2± 0.3

45 9210± 2 2065± 2 138.2± 0.5

50 11050± 4 2509± 3 166.0± 0.5

55 12660± 4 2962± 4 196.6± 0.5

Table 4.1: Detected photons with different power settings. The pump power on thecrystal does not coincide with the optical power emitted by the diode laser, much ofwhich is lost at the first fiber coupling and some is reflected by the optical componentsbefore the interferometer. The third column lists the average tally of the two SPADs.

For the rest of the experiment we set the laser diode at a driving current of 50 mA,corresponding to around 11 mW of pump light entering the interferometer. With theaforementioned values of ηSPAD and ηcoupling, we can estimate that approximately onein a billion pump photons is converted and around 3.5 million pairs are produced eachsecond. Considering that the measurement apparatus normally loses 95 − 99 % of thesignal (depending on the settings, see Section 4.2), the rate of coincident counts used inthe actual experiments is in the low thousands per second.

Another necessary condition for the success of the tomography process is that the mea-sured state remains the same for all its duration. We could confirm this by means ofchecking the pump power in the two arms of the interferometer, which is connected tothe a and b coefficients of the previous section.

The Experiment 41

101 102 103 104

Pump power ( W, logscale)

103

104

105

106

Coun

ts (l

ogsc

ale)

Average of the two channelsCoincidences

Figure 4.3: Counts as a function of power with linear fit of the logarithmic curve.Error bars are smaller than marker size. The linear law is confirmed by slopes ≈ 0.95.

0 50 100 150 200 250 300 350Time (minutes)

3.2

3.4

3.6

3.8

4.0

4.2

Pum

p Po

wer (

mW

)

ClockwiseCounterclockwiseAverage

21.2

21.3

21.4

21.5

21.6

21.7

21.8

Tem

pera

ture

(°C)

Temperature

0 50 100 150 200 250 300 350Time (minutes)

0.004

0.002

0.000

0.002

0.004

0.006

0.008

Rela

tive

varia

tion

of th

e ra

tio

Rel. Var. Ratio21.2

21.3

21.4

21.5

21.6

21.7

21.8

Tem

pera

ture

(°C)

Temperature

Figure 4.4: Power in the clockwise(blue) and counterclockwise (red)arms as a function of elapsed timefrom the start of the acquisition.Their average is shown in grey. Theblack curve is a concurrent measure

of temperature.

Figure 4.5: Variation of the ratiobetween powers in the two arms, rela-tive to its initial value. Temperature

in black.

Figure 4.5 shows that the ratio between the two values is approximately constant. Thismeans that the SM fiber that brings the pump light to the Sagnac interferometer doesnot induce time dependent transformations on the polarization state (it could change thephase φ between the horizontal and vertical components, but these unitaries form a set ofmeasure 0 in the space of all possibilities, therefore this behavior is extremely unlikely).Supposing that the exit fibers are similar and that all losses are polarization-independentor anyway also constant, we can conclude that the state entering the measurementapparatus is stable for long periods of time.

4.1.4 Temporal Coherence of the Photons

Temporal coherence is a measure of the phase correlations between two values of awave that are separated in time and shows how strongly they can interfere with eachother. Since interference is the foundation of our measurement setup it is important tounderstand if and how well the used light can display this phenomenon.

42 The Experiment

SM Fiber

Source

Polarizer

Polarizer

Linear stageBS

SPAD

SM Fiber

Figure 4.6: Scheme of the Michelson interferometer used to study temporal coherence.

In order to study this property we built a Michelson interferometer with a moving arm(Figure 4.6) and sent the single photons produced by our source through it. After takingone of the paths, photons reenter the beam splitter and reach a single mode fiber wherethe two spatial modes are projected onto one. In this way the strength of the interferencedoes not depend on their spatial overlap but only on how well they are coupled into thefiber and on their optical path difference ∆l. The intensity, measured as photon countsper second by the SPAD, can be expressed as:

Iint(∆l) = IF + IV (∆l) +2√IF IV (∆l)

IF + IV (∆l)γ cos

(ϕ(∆l)

)(4.13)

where IF and IV (∆l) are the intensities of the beams taking the path with fixed orvariable length, ϕ is the phase difference between the two and γ ∈ [0, 1] is a coefficientrepresenting the phase correlation. Due to the precision of the linear stage that movesthe mirror, we could not observe the oscillating part of the pattern but only its envelope,which is, however, enough to infer meaningful information about temporal coherence.

We changed the length of the moving arm with steps of 1 µm, each corresponding to anactual variation of ∆l of 2 µm because light travels its path twice before getting backto the beam splitter. After measuring IF once and for all, we registered Iint(∆l) andIV (∆l) at each position, considering only coincidence counts in the two SPADs, so asnot to include background light entering the fiber. Then, we measured the visibility as

V ≡ Iint,max − Iint,minIint,max + Iint,min

(4.14)

where max and min refer to intervals of ten steps. If we can consider IV (∆l) constant

in each 20 µm wide interval, a reasonable approximation, then V ≈ 2√IF IV (∆l)

IF+IV (∆l) γ, fromwhich we can extract γ. By defining the coherence length ∆lc as the value of ∆l forwhich the phase correlation is 1/e, Figure 4.7 tells us that

∆lc ≈ (9± 1) · 10−5 m (4.15)

The Experiment 43

0.2 0.1 0.0 0.1 0.2 0.3l (mm)

0

2000

4000

6000

8000

10000

12000

14000

Coun

ts p

er se

cond

Counts/s 1.00

0.75

0.50

0.25

0.00

0.25

0.50

0.75

1.00

Coef

ficie

nt

Figure 4.7: Results of the coherence analysis. Due to imperfect alignment, IV changeswith ∆l, although it can be considered constant in each 20 µm wide interval. This causesthe slight asymmetry in the power values (blue) between positive and negative OPL

regions. This has been taken into account by correcting γ (red) as γ = V · IF +IV (∆l)

2√IF IV (∆l)

.

The gray line corresponds to γ = 1/e.

which corresponds to a temporal coherence of ∆tc = ∆lcc ≈ (30 ± 3) · 10−14 s. This

small value shows that a strong interference is only available if paths are very carefullybalanced. The presence of lateral lobes in the visibility graph suggests that the spectrumof the analyzed photons might be multi-modal.

4.2 The Measurement System

The center of this experiment is the measurement setup. As briefly mentioned at thebeginning of this chapter, this apparatus has the purpose of faithfully implementingthe direct measurement strategies of Section 3.2 to the polarization states of photons.Specifically, it was built with the DRDO scheme in mind, but can also easily performall the steps of the simpler DRDD protocol. First we built a version that is only ableto actualize strong couplings between object and ancilla, then we made some additionsthat allowed to adjust the weakness of the interaction, which will be described in Section4.3.

4.2.1 Design

The setup was developed around three goals:

• Separation between object and ancilla.

• Measurements on the ancilla.

• Postselection.

44 The Experiment

Since we chose the path as our pointer, the first objective can be fulfilled by any opticaldevice that can spatially separate a beam into two polarized components. One of thetwo states in the measurement basis {|H〉, |V 〉} is chosen as |aj〉 and sent into path |1A〉while the other stays in |0A〉. Considering for simplicity a pure initial state, the actionof this piece can be summarized by

|ψ〉 = (cj⊥|aj⊥〉+ cj |aj〉)⊗ |0A〉 → |ψ′〉 = e−iπ2

Πaj⊗σyA |ψ〉 = cj⊥|aj⊥〉|0A〉+ cj |aj〉|1A〉(4.16)

Notice that |0A〉 labels both the beam course before the displacement and one of thepaths after it. The strength of the interaction is maximal (θ = π

2 ), which means that thetwo polarizations are completely separated into two orthogonal (non-overlapping, per-fectly distinguishable) pointer states and there is symmetry between object and ancilla.

From formulas (3.6), (3.7) and (3.9), it is clear that the setup must projectively measureσx, σy and Π1 on the ancilla, therefore the pointer state can collapse on one of sixpossibilities: |±A〉 (eigenstates of σx), |± iA〉 (eigenstates of σy), |1A〉 or |0A〉 (successfulor unsuccessful projection on |1A〉). Simultaneously, the object collapses too, so thatthe joint system lies in one of the six separable states of Table 4.2.

Measurement Outcome Probability Final state

σx 1 12 (cj⊥|aj⊥〉+ cj |aj〉)⊗ |+A〉

σx -1 12 (cj⊥|aj⊥〉 − cj |aj〉)⊗ |−A〉

σy 1 12 (cj⊥|aj⊥〉 − icj |aj〉)⊗ |+ iA〉

σy -1 12 (cj⊥|aj⊥〉+ icj |aj〉)⊗ | − iA〉

Π1 1 |cj |2 |aj〉 ⊗ |1A〉Π1 0 |cj⊥|2 |aj⊥〉 ⊗ |0A〉

Table 4.2: The states after the useful pointer measurements. These results are alsovalid for the second coupling provided aj and aj⊥ are replaced by b0 and b1.

We implemented the measurement process by closing the interferometer in a way thatcan morph |ψ′〉 selectively into each of these states. In the first four cases a phase shiftmust be introduced between the two beams that have then to be recombined so that halfthe power is discarded, whereas for the measurements of Π1 one arm has to be blocked.

Subsequently, another identical block receives the photons, realizes the coupling withthe second ancilla and performs the same measurements on it. The only difference withthe first one is that the interacting polarization states are |b0〉 = |D〉 = |H〉+|V 〉√

2and

|b1〉 = |A〉 = |H〉−|V 〉√2

instead of being |aj〉, |aj⊥〉 ∈ {|H〉, |V 〉}.

Measurement Outcome Delay Block

σx 1 0 rad /

σx -1 π rad /

σy 1 3π2 rad /

σy -1 π2 rad /

Π1 1 Any |0A〉Π1 0 Any |1A〉

Table 4.3: Phase delays that must be applied to the |1A〉 path compared to |0A〉.Notice that a delay of 3π

2 rad to a beam is equivalent to a delay of π2 rad to the other.

The Experiment 45

Finally, postselection on |ak〉 ∈ {|H〉, |V 〉} is carried out on the object system and theresults are sent to a detector that counts the photons for a fixed exposure time.

4.2.2 Practical Realization

PBD Shutters LCR HWP PBD

Figure 4.8: Scheme of each Mach-Zehnder interferometer.

The key building blocks of this apparatus are the two identical Mach-Zehnder inter-ferometers that split and recombine the two polarizations and carry out the ancillameasurements (Figure 4.8). These are based on PBDs, provided by Thorlabs undermodel name BD40, which employ birefringent calcite crystals to separate light into twoparallel beams at a distance of approximately (4.1 ± 0.1) mm at the operating wave-length of 809 nm. We chose to use PBDs instead of more conventional beam splittersbecause they made the alignment process much easier and allowed us to interact withboth beams using the same optics, thus reducing the cost of the setup.

Various possibilities were considered for the application of the phase delays between thepaths. The most straightforward one was the use of two glass plates, one of which couldbe tilted so as to tune the optical path length (OPL) of the photons passing throughit. The other plate was necessary because of the short coherence time of our source,which did not allow for too big a difference in OPL. This option was discarded when wenoticed that imperfections in the available glass plates caused a deflection of the beamswhich was difficult to compensate for, especially considering that the plates had to befrequently moved during each measurement.

Then we contemplated tiltable birefringent waveplates spanning over both beams. Ifaligned with their optical axes parallel or perpendicular to the polarization of the in-coming light, a rotation around said axes could provide the desired phase delay. Howeverthe plates had to be tilted at an angle so great that the beams were clipped, thus forcinganother change of strategy

Finally, our choice fell on liquid crystal retarders (LCRs), devices which can change theirbirefringence with an applied voltage. Unfortunately, the only available model, ThorlabsLCC1221-A, is optimized for a shorter wavelength and can only provide a phase shift inrange [∼ 0,∼ 0.8π] rad at 809 nm, therefore we mounted the two pieces on motorizedrotators (Thorlabs PRM1Z8) so as to change the orientation of their optical axes. Inthis way we effectively doubled the achievable delay range to [∼ −0.8π,∼ 0.8π] rad. Weintroduced an additional fixed phase shift of ∼ 0.8π rad by slightly tilting the PBDs tocover the more useful interval [∼ 0,∼ 1.6π] rad, as prescribed by Table 4.3.

46 The Experiment

After each LCR, a HWP with its axis oriented at π8 rad compared to the horizontal has

the purpose of changing the polarizations so that each of the exits of the subsequent PBDcollects half the power of each of the incoming beams, as required by the measurementsof σx and σy. For those of Π1 we blocked the |0〉 path (which is the one taken by the|aj⊥〉 and |b1〉 polarizations) using manual shutters. In this case the halving is undesired,therefore we compensated for it by doubling the resulting counts in post-processing.

We chose to position the closing PBD of each interferometer in the same manner asthe opening one, so that both arms have the same OPL. This, again, is necessary dueto the short coherence time of photons, but introduces another problem: compared tothose of Table 4.2, the final states are polarized symmetrically with respect to the Daxis after the first interferometer and to the H axis in the second. We took this intoaccount in the design of the second interferometer, which is a duplicate of the first onebut is rotated by π

4 rad along the propagation direction, so as to distinguish between|D〉 and |A〉 polarizations. At the end of it, we added a HWP which not only providesthe needed correction, but also chooses the postselected state that passes through thefinal polarizer and reaches the detector.

In order to implement the DRDD protocol, that does not need two ancillae, this lastHWP can be oriented so as to disregard the contribution of one of the two paths in thesecond interferometer, that thus acts only as a glorified postselecting polarizer.

By considering all the useful combinations of measurement outcomes in the two pointersand possible choices of |aj〉 and |ak〉, we can measure photon counts that are proportionalto the desired joint probabilities that appear in the mean values of equations (3.6), (3.7)

and (3.9) (and include factor Tr(Πak %)). For instance if 〈σ(aj)yA σ

(b0)yB 〉ak is needed to find

element HV of the density operator, four acquisitions must take place, one for eachprojector of the spectral decomposition of the observable σyAσyB. According to Table4.3, the first interferometer is set to transmit a phase delay of 3π

2 to the |H〉 polarization,and the second does the same to the |D〉 polarization, so that after postselection on |V 〉count Ny(+1),y(+1),HV is recorded. Similar combinations of phases are set for the otherthree counts. Then

〈σyAσyB〉′ ∝∑

a,b=±1

abNya,yb,HV (4.17)

If normalization is needed, it can be carried out by setting the two interferometers at 0delay and by measuring the |H〉 and |V 〉 components of the exiting state. The sum ofthese two counts, corrected because of the unnecessary halving operated by the HWPs,is a measure of the total number of photons that can reach the detectors and is theproportionality constant between data and probabilities.

After the second interferometer, a QWP helps performing the measurements needed bythe QST protocol, which was also put to the test. In this case the rest of the setup is setin such a way that the transformation applied to the state is known and can be takeninto account during the analysis of the results.

The Experiment 47

PBD

Shutters

LCR

HWP

Polarizer

SPAD

Source

First

Coupling

Postselection and Detection

PBD

QWP

SPAD

HWP

SM Fiber

SM Fiber

PBD

Shutters

LCR

HWP

Second

Coupling

PBD

Figure 4.9: Scheme of the entire measurement system.

48 The Experiment

4.2.3 Phase Calibration and Stability

The characterization of the LCRs is a fundamental preliminary phase of the experimentas it allows to achieve fine control over the phase delay between different polarizationcomponents. First we directly measured the retardation vs voltage curve using a simplesetup: we placed each device between two polarizers and added a rotating HWP. Forease of explanation, suppose that the first polarizer selects |A〉 and the second |D〉, thenif the slow axis of the LCR is horizontal and the HWP has its fast axis oriented at angleα with respect to it, the power at the detector is

P (α) ∝ 1 + cos(Γ) cos(4α) (4.18)

where Γ is the phase delay. For different voltage settings we scanned α and fitted thisequation to the resulting power curve, thus finding cos(Γ) and ultimately Γ itself.

Laser source Polarizer |A⟩ LCR Rotating HWP

Polarizer |D⟩ PowerMeter

Figure 4.10: Scheme of this measurement setup.

0 5 10 15 20 25Applied Voltage (V)

0.0

0.5

1.0

1.5

2.0

2.5

Phas

e sh

ift (r

ad)

LCR #1LCR #2

Figure 4.11: Retardation curve as a functionof voltage. Measured with λ ∼ 808 nm laser

light.

We assumed Γ to be positive due to thedenomination of fast and slow axis (whichare indistinguishable with these measure-ments, therefore we had to rely on themarkings on the device), however, a signerror is not particularly relevant as it canbe corrected with an arbitrary redefinitionof |H〉 and |V 〉. The fact that neitherof the two LCRs can transmit a delay of3π2 (Figure 4.11) justified their aforemen-

tioned installation on rotators, which dou-bled their range by switching the roles ofthe axes.

We also studied the behavior of these de-vices in the true experimental conditions,that is, inside the interferometers. Bysending light into each of them with a

known linear polarization and by measuring the same component at the exit, we recordedpower (counts per second when using single photons) at different voltage settings.Through parabolic regressions we found the voltages corresponding to maximum andminimum power, which indicate 0 and π rad delay respectively. Linear interpolationsof the curves around the points near half the maximum power allowed to retrieve the

The Experiment 49

0 5 10 15 20 25Applied Voltage (V)

0

1000

2000

3000

4000

Powe

r (Co

unts

/s)

Pos2Pos1

Figure 4.12: Power as a function of voltage. The red and blue curves refer to the twodifferent orientations of the optical axis of the LCR (parallel to |H〉 or |V 〉 in the first

interferometer, to |D〉 or |A〉 in the second).

voltages needed to transmit π2 or 3π

2 rad. These values include not only the action ofthe LCR but also the phase due to OPL imbalances in the rest of the interferometer.Figure 4.12 also shows that visibility exceeds 98% which is optimal for the experiment.

This procedure had to be repeated before each measurement to find the precise value ofthe voltage, which can change over long periods of time. We studied this phenomenonby acquiring curves like 4.12 in sequence and measuring the variation of phase delay atfixed voltage. Figure 4.13 shows that the interferometer is stable for at least a couple ofhours after calibration and its changes are correlated to the temperature of the room.The fact that the sign of the phase variation is the same in the two curves suggeststhat the culprit is not the LCR but rather the PBDs or more precisely their supports.Indeed the relative inclination of the two PBDs in each interferometer can change overtime if temperature variations cause their mounts to move. Due to the macroscopiclength of the birefringent material, even a small angle can induce relevant changes inthe geometric path and therefore in the phase delay. This has justified the use of precisiontilters (Thorlabs Polaris K1S5) which have demonstrated better stability compared toour initial trials with cheaper models.

In conclusion, provided the appropriate mounts are used and the room temperature doesnot change too much, this setup guarantees enough stability for the experiment.

50 The Experiment

0 100 200 300 400 500 600 700Time (minutes)

0.025

0.000

0.025

0.050

0.075

0.100

0.125

0.150

Phas

e Va

riatio

n (ra

d)

Near MaxNear Min

27.175

27.200

27.225

27.250

27.275

27.300

27.325

27.350

Tem

pera

ture

(°C)

Temp

Figure 4.13: Variation of phase delay over time, relative to arbitrary initial 0. Theblack curve is a concurrent measure of temperature.

4.2.4 Analysis of the Statistical Error

In order to provide an estimate for the statistical error to attribute to each count, wetook samples of repeated acquisitions in the same conditions and performed a simpleanalysis to extract mean and standard deviation. We used coincident counts across thetwo SPADs as in the actual experiment and covered a wide range of exposure times.We did this in four different situations, one in which both interferometers were inactive(“II”), meaning that one arm was blocked so that light could pass through the other butwithout any interference effect, one in which both were active (“AA”), meaning thatinterference took place after both of them, and two in which one was active and one wasnot (“IA” and “AI, where the order of symbols is that of the two interferometers).

102 103 104

Mean in repeated acquisitions (counts, logscale)

101

102

Stan

dard

dev

iatio

n in

repe

ated

acq

uisit

ions

(cou

nts,

logs

cale

)

AAAIIAII

Figure 4.14: Relation between standard deviation and mean with linear fit of thelogarithmic curve.

The Experiment 51

Figure 4.14 shows that the poissonian law is respected in all configurations. We verifiedthis through linear regressions on the logarithmic curves which provided slopes that arevery close to 1

2 (min = 0.503±0.007, max = 0.517±0.004). In other words, the observedstandard deviation is well approximated by the square root of the average.

4.3 Weakening the Coupling

In order to evaluate the protocols at different degrees of strength, we made a few mod-ifications to the measurement setup. The most fundamental one was the addition of aHWP in one of the two arms of each interferometer: this new piece takes the role ofimplementing the coupling between object and ancilla, which is no longer properly thepath taken by the photons.

Remaining true to the theoretical procedure, one of the two polarization components|H〉 or |V 〉 is selected as |aj〉 by the first PBD and encounters the subsequent HWP. Hereit gets rotated by angle θA which represents the strength: when θA = π

2 , |aj〉 becomes|aj⊥〉 so that both arms are polarized in the same way. It may be said that inside theinterferometer, the polarization acts as the pointer and the separation in two paths isnecessary only because it allows to interact selectively with one component.

StandardManualRotator

Plate

3D Printed support

Beam can pass:- through the plate- next to the plate

Figure 4.15: Photograph of the custommount.

The HWP is about 2 mm thick and there-fore introduces a difference in OPL that ismuch greater than the coherence length ofour photons. In order to compensate, wealso added a plate to the other arm, whichwas oriented in a way that did not changethe incoming polarization. These two op-tics were mounted on custom 3D printedsupports (Figure 4.15) designed by LucaCalderaro specifically for this experimentthat allowed one beam to pass unperturbednext to the plate and could be placed in-side standard 1′′ manual rotators (ThorlabsCRM1). We used the Michelson interfer-ometer of Subsection 4.1.4 to measure thatthe remaining OPL imbalance was approx-

imately 10 µm, which we could correct by tilting the second PBD. Another HWP wasinserted at the beginning of the setup to switch |aj〉 between |H〉 and |V 〉 withoutphysically moving pieces in the two arms.

The ancilla measurements are performed by a rotating QWP and HWP placed insidethe interferometer that are hit by both paths. At the exit, a LCR helps to tune therelative phase between the arms so that it is maintained at a known value during theentire experiment.

The second interferometer is a twin of the first, except for the fact that it is rotated byπ4 along the propagation direction and does not have any HWP before the PBD as |b0〉is always |D〉 and does not have to be switched to |A〉. Postselection is implemented bya HWP and a PBS just before the detector.

52 The Experiment

With an appropriate set of orientations of the plates, we were able to apply the necessarypointer projections as prescribed by the protocols. Again, the second interferometer wasused as a polarizer (i.e. it had only one active arm) for the DRDD scheme, whereas QSTwas realized using the waveplates of the first interferometer at maximum strength.

|aj〉 Measurement Outcome QWP HWP Block

|H〉 σx 1 3π4 rad 3π

8 rad /

|H〉 σx -1 π4 rad π

8 rad /

|H〉 σy 1 π2 rad 3π

8 rad /

|H〉 σy -1 π2 rad π

8 rad /

|H〉 Π1 1 0 rad 0 rad |0A〉|H〉 Π1 0 0 rad 0 rad |1A〉|V 〉 σx 1 π

4 rad 3π8 rad /

|V 〉 σx -1 3π4 rad π

8 rad /

|V 〉 σy 1 π2 rad π

8 rad /

|V 〉 σy -1 π2 rad 3π

8 rad /

|V 〉 Π1 1 0 rad 0 rad |0A〉|V 〉 Π1 0 0 rad 0 rad |1A〉

Table 4.4: Positions of the fast axes of the plates that carry out the measurements inthe first interferometer. All angles are measured with respect to the |H〉 axis. These

values can change if the relative phase between the paths is set differently.

|aj〉 Measurement Outcome QWP HWP Block

|H〉 σx 1 π2 rad π

8 rad /

|H〉 σx -1 0 rad 3π8 rad /

|H〉 σy 1 π4 rad 3π

8 rad /

|H〉 σy -1 π4 rad π

8 rad /

|H〉 Π1 1 π4 rad 3π

4 rad |0A〉|H〉 Π1 0 π

4 rad 3π4 rad |1A〉

|V 〉 σx 1 0 rad π8 rad /

|V 〉 σx -1 π2 rad 3π

8 rad /

|V 〉 σy 1 π4 rad π

8 rad /

|V 〉 σy -1 π4 rad 3π

8 rad /

|V 〉 Π1 1 π4 rad 3π

4 rad |0A〉|V 〉 Π1 0 π

4 rad 3π4 rad |1A〉

Table 4.5: Positions of the fast axes of the plates that carry out the measurements inthe second interferometer. All angles are measured with respect to the |H〉 axis. These

values can change if the relative phase between the paths is set differently.

The Experiment 53

PBD

Shutters

HWP

Polarizer

Source

First

Coupling

Postselection

and Detection

PBD

SPAD

HWP

Strength Plate

QWP

Compensation Plate

LCR

PBD

Shutters

HWP

Second

Coupling

PBD

Strength Plate

QWP

Compensation Plate

LCR

HWP

SM Fiber

SPAD

SM Fiber

Figure 4.16: Scheme of the entire measurement system.

Chapter 5

Results

The main experiment consisted of two phases: the first was aimed at verifying thevalidity of our protocols at full strength and used the setup of Section 4.2, whereas thesecond focused on the role of the coupling strength and followed the scheme of Section4.3. In both cases we tested all the methods at our disposal in order to compare themeffectively. We calculated unnormalized density matrices %U from raw counts in thefollowing different ways:

• DRDO protocol. From expression (3.7):

%Ujj =d2

sin2 θA sin2 θBNz(−1),z(−1),jk ∀k

<(%Ujk) = − d

2 sin θA sin θB

∑a,b=±1

abNya,yb,jk j 6= k

=(%Ujk) =d

2 sin θA sin θB

∑a,b

abNxa,yb,jk j 6= k

(5.1)

Despite it being unnecessary to measure both non-diagonal elements separately, wechose to do it as well in order to prove the ability of this scheme to pick any matrixelement individually. Although these formulas are exact, %U is not guaranteed tobe hermitian due to experimental errors. Its diagonal elements and trace are,however, necessarily real. This scheme employs 18 different measurements.

• DRDO Lundeen protocol. From expression (2.33):

<(%Ujk) =d

4 sin(θA) sin(θB)

∑a,b

ab(Nxa,xb,jk −Nya,yb,jk)

=(%Ujk) =d

4 sin(θA) sin(θB)

∑a,b

ab(Nya,xb,jk +Nxa,yb,jk)

(5.2)

We tested this method even when the approximations on which it is based are notvalid (i.e. when the coupling is not weak) to better understand its limitations.Just as above, %U is not guaranteed to be hermitian but this time the diagonalelements can be complex. Even in absence of experimental errors, hermiticity is nota necessity unless θA,B is small. This scheme employs 64 different measurements.

55

56 Results

• DRDO Lundeen protocol with corrections. From expression (3.6):

<(%Ujk) =d

4 sin θA sin θB

(∑a,b

ab(Nxa,xb,jk −Nya,yb,jk)

+ 2 tanθB2

∑a

aNxa,z(−1),jk + 2 tanθA2

∑b

bNz(−1),xb,jk

+ 4 tanθA2

tanθB2Nz(−1),z(−1),jk

)

=(%Ujk) =d

4 sin θA sin θB

∑a,b

ab(Nya,xb,jk +Nxa,yb,jk) + 2 tanθB2

∑a

aNya,z(−1),jk

(5.3)

Again this matrix is not necessarily hermitian, although it would be in absence ofexperimental errors and in any conditions of strength. This scheme employs 92different measurements.

• DRDD protocol. From expressions (3.9) and (3.11):

ρUA1(jl) =1

2 sin(θA)

∑a

a(Nxa,jl + iNya,jl)

ρUA2(jl) =1

sin(θA)Nz(−1),jl

%Ujk = d tan

(θA2

)δjkρ

UA2(jl) +

∑l

e2πil(j−k)

d ρUA1(jl)

(5.4)

No guarantee of hermiticity in presence of experimental errors. This scheme em-ploys 20 different measurements.

• DRDD Lundeen protocol. From expressions (2.37) and (2.35):

ρUA1(jl) =1

2 sin(θA)

∑a

a(Nxa,jl + iNya,jl)

%Ujk =∑l

e2πil(j−k)

d ρUA1(jl)(5.5)

No guarantee of hermiticity in presence of experimental errors or in their absenceif θA is not small enough. This scheme employs 16 different measurements.

• QST. Due to the small dimension of the system we could easily perform a standardtomography using the projectors on |H〉, |V 〉, |D〉 and |R〉. From the correspondingcounts we found

%UHH = NH

%UV V = NV

<(%UHV ) = <(%UV H) = ND −1

2(NH +NV )

=(%UHV ) = −=(%UV H) = NR −1

2(NH +NV )

(5.6)

This matrix is hermitian and is measured with only 4 acquisitions.

Results 57

Because many of the needed projections are shared between the protocols, the totaltally of acquisitions is 116. The reconstruction of a state took approximately 17 minutesin the first experiment and 34 in the second, an extension mostly due to the increasednumber of rotations of the plates. In both cases the exposure time was set at 2 secondsso that the value of all counts was in the thousands (except for those proportional to a〈Π1A,B〉 which were much smaller in case of low coupling strength).

Although it would have been possible to normalize each matrix element for exampleusing NH +NV as the constant, we followed the steps that we imagine would be chosenin a tomography experiment: we took the hermitian part of each matrix and normalizedit with its trace:

%U → %H =%U + %U†

2→ %N =

%H

Tr(%H)(5.7)

The resulting matrices are all hermitian and all have unit trace, although they are notnecessarily positive semidefinite and therefore may not be physical density operators.However, in order to compare the methods in fair conditions, we did not add a likelihoodmaximization step and used %N as our starting points in the analysis I shall present caseby case in the next sections.

5.1 Results at Maximum Strength

Our source is capable of sending to the measurement apparatus both mixed and (almost)pure states. In particular, the rotating HWP before the Sagnac interferometer canchange the balance of pump light in the two arms so that if one of them is completelyinactive, the entirety of the photons reaching one exit is |H〉 polarized while those thatarrive at the other are |V 〉 polarized. This means that despite the uncontrollable actionof the two fibers, the state entering the measurement system is approximately pure. Onthe other hand, when the crystal is hit by both arms, each fiber will collect both |H〉and |V 〉 photons, so that the state at its exit is mixed (completely mixed when the twopolarizations are perfectly balanced inside the fiber).

We used this technique to generate states with different degrees of purity, which wemeasured as the trace of the square of the matrix resulting from the QST method. Thedata confirm that we were able to produce a state very close to being completely mixed,indeed its purity is 0.501 ± 0.003, while the theoretical limit is Tr(%2) = 1

d = 0.5. Themaximum purity we observed is only 0.936 ± 0.006 due to the fact that even with thehelp of a QWP before the Sagnac interferometer, the pump light entering it was notlinearly polarized. Moreover, imperfections in the PBS caused imbalanced extinctionratios in the two arms which explain the different heights of the two peaks in Figure 5.1.

Since the polarization of the pump light after the fiber did not change during the ex-periment, the measured states lie on a straight line in the Bloch sphere, as is shown inFigure 5.2 which plots the QST results.

58 Results

0 10 20 30 40Angle of the HWP (deg)

0.5

0.6

0.7

0.8

0.9

Purit

y

|D |L

|H

Figure 5.1: Purity as Tr((%NQST )2)as a function of the angle of the rotat-ing HWP placed before the Sagnacinterferometer, measured in respect

of an arbitrary 0.

Figure 5.2: Visualization of theQST results on the Bloch sphere.

Then we calculated %N using the previously introduced formulas from the various meth-ods and θA = θB = π

2 . A first comparison can be done by simply observing thatthe estimated density operators are similar. From Figure 5.3 it is clear that all themeasurement protocols are in agreement, only the methods that are based on invalidapproximations (DRDD Lundeen and DRDO Lundeen) are off the mark.

|D |L

|H

DRDODRDO LundeenDRDO Lundeen + corrections

DRDDDRDD LundeenQST

(a) Purest state.

|D |L

|H


DRDDDRDD LundeenQST

(b) Most mixed state.

Figure 5.3: Visualization of two measured states. Each arrow corresponds to theresult of one of the protocols.

Results 59

To express this in more quantitative terms, we measured the trace distance between theresult of each scheme and that of the QST, treated as reference.

t(%N , %NQST ) =1

2Tr√

(%N − %NQST )†(%N − %NQST ) =1

2

√ ∑i=x,y,z

(ri − ri,QST )2 (5.8)

in which ri are the coefficients of the Bloch vector representing the density matrix.

0.5 0.6 0.7 0.8 0.9Purity

0.0

0.2

0.4

0.6

0.8

1.0

Tra

ce d

ista

nce

t

DRDODRDO LundeenDRDO Lundeen + correctionsDRDD

Figure 5.4: Trace distances between each method and QST. Error bars are calculatedusing a simulation of the experiment in which 104 samples were generated from aPoissonian distribution with the actual measurement results as mean (Appendix A.5.2).

Figure 5.4 shows that for our methods t < 0.15, while the DRDO Lundeen protocolalways fails to produce accurate results. The trace distance relative to the DRDDLundeen protocol is not plotted as it is always greater than 40. This is because atmaximum strength all the information about the diagonal elements is recorded in ρA2,therefore an estimate that only considers ρA1 necessarily returns values close to 0. Whennormalization occurs, non-diagonal elements skyrocket and the matrix is non-physical.It is important to remember than if t > 1, the matrix is necessarily out of the Blochsphere and is often completely unrelated to the true state, indicating that the methodhas utterly failed to provide any meaningful result.

In order to prove the validity of the protocols at higher purities, we added a polarizer, aQWP and a HWP at the beginning of the measurement apparatus so as to prepare thesix states |A〉, |D〉, |H〉, |L〉, |R〉, |V 〉. All the purities obtained from the QST matriceslie in interval [0.94, 0.99]. This time we calculated the trace distances with respect to thetheoretical density operators, and again all the methods that are expected to succeedreport acceptable values, while Lundeen’s protocols generally fare worse, with t > 0.5 for

60 Results

the DRDO and t > 5 for the DRDD protocol. For this reason Figure 5.6 does not showany of the DRDD Lundeen results nor two of the DRDO ones which lie well outside theBloch sphere.

|D|L

|H


DRDDDRDD LundeenQST

(a) A linear polarization.

|D |L

|H


DRDDDRDD LundeenQST

(b) A circular polarization.

Figure 5.5: Visualization of two measured states. Each arrow corresponds to theresult of one of the protocols.

A D H L R VState

0.0

0.2

0.4

0.6

0.8

1.0

Tra

ce d

ista

nce

t

DRDODRDO LundeenDRDO Lundeen + correctionsDRDD

Figure 5.6: Trace distances between each method and the theoretical density matrix.Two points of the DRDO Lundeen protocol are not shown because they are above 5.

Results 61

These results prove that the DRDO protocol is capable of measuring accurate densityoperators using maximally strong couplings. Trace distances are generally satisfactoryregardless of the purity of the states. The fact that they are often incompatible with 0is due to the complexity of the apparatus, which, despite the fine alignment, can stillintroduce systematic errors. It is also clear that the schemes that are based on weakmeasurements are not functional with strong ones, especially in the DRDD case.

5.2 Results for Variable Strength

After applying to the measurement setup the modifications described in Section 4.3,we proceeded with the second part of the experiment. Due to the increased number ofrotations (both manual and motorized) required by the implementation, we evaluatedonly two states, one mixed and one (almost) pure, each with five different values ofstrength θ ≡ θA = θB. We measured these values using the angles of the rotators thatcontrol the HWPs which take care of the coupling, or more precisely twice the differencebetween such angles and those corresponding to orientations of the plates that do notchange the polarization of the incoming light. Although inaccurate, this estimate issurely sufficient to discriminate the trials between strong and weak regime.

|D |L

|H

Figure 5.7: The five QST results for themixed state.

In order to produce the mixed state we madesure that the two arms of the Sagnac interfer-ometer of the source were approximately bal-anced in terms of pump power. Figure 5.7shows the results of the QST, which we re-peated after each trial although it is indepen-dent of the strength. This visualization con-firms that during the entire experiment the in-coming polarization did not change, with thehighest trace distance between the resultingmatrices being only 0.02. Moreover the statestayed close to being maximally mixed, withpurities always smaller than 0.51.

Figure 5.8b again confirms that the methodsthat are based on the weak approximation donot work properly at higher values of strength.Figure 5.8a instead shows that for small θ theDRDD Lundeen scheme can produce accurate

results. It is interesting to observe that the DRDO Lundeen protocol, with or withoutcorrections, gives a higher trace distance than expected (Figure 5.9). We ascribe this toslight misalignments in the setup which can introduce systematic errors in the countsthat, however small, become important at low strength (as explained in Subsection3.3.2).

62 Results

|D |L

|H


DRDDDRDD LundeenQST

(a) Measurements with θ ≈ 0.07 rad.

|D |L

|H


DRDDDRDD LundeenQST

(b) Measurements with θ ≈ π2rad.

Figure 5.8: Visualization of the results of two measurements of the mixed state. Eacharrow corresponds to one of the protocols.

0.0 0.2 0.4 0.6 0.8 1.0(units of /2 rad)

0.0

0.2

0.4

0.6

0.8

1.0

Tra

ce d

ista

nce

t

DRDODRDO LundeenDRDO Lundeen + correctionsDRDDDRDD Lundeen

Figure 5.9: Trace distances between each method and QST.

Results 63

|D |L

|H

Figure 5.10: The five QST results for thepure state.

These problems become more evident in thecase of the pure state, which we generatedthrough the insertion of a linear polarizer atthe beginning of the measurement setup. TheQST results are close to each other (the maxi-mum mutual trace distance is 0.06) but do notcoincide with the expected state |D〉. Indeed asmall rotation towards the |H〉 pole is notice-able in Figure 5.10, which suggests again thepresence of imperfections in the setup.

The conclusions we can draw from these re-sults are similar to the previous, although theerrors of Lundeen’s schemes are more relevant.Moreover, the vertical bars of the green plotof Figure 5.9 show the higher relative statisti-cal uncertainties of the DRDO scheme, whichare due to small factor sin4(θ) at low strength.

The differences between Figures 5.12 and 5.9 also highlight the fact that the quality ofthe results can vary with the state. However it is clear that the choice of operating inweak conditions is generally inconvenient even if the appropriate measurement protocolsare used.

|D |L

|H


DRDDDRDD LundeenQST

(a) Measurements with θ ≈ 0.07 rad.

|D |L

|H


DRDDDRDD LundeenQST

(b) Measurements with θ ≈ π2rad.

Figure 5.11: Visualization of the results of two measurements of the pure state. Eacharrow corresponds to one of the protocols.

64 Results

0.0 0.2 0.4 0.6 0.8 1.0(units of /2 rad)

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Tra

ce d

ista

nce

t

DRDODRDO LundeenDRDO Lundeen + correctionsDRDDDRDD Lundeen

Figure 5.12: Trace distances between each method and the expected matrix |D〉〈D|.The point of the DRDD Lundeen protocol at maximum strength is not plotted because

it is too far apart from the others (4.9± 0.3).

Chapter 6

Conclusions

After the detailed study of the fundamentals of quantum mechanics and measurementtheory, this thesis has investigated the new possibilities that are offered by weak valuesespecially in the field of quantum state reconstruction (Chapter 2, Reference [3]). Despitethe revolutionary contribution of the work of Aharonov, Albert and Vaidman [20] toquantum sensors and fundamental research (Section 2.3, References [26, 38, 39]), werecognized that weak measurements are not strictly necessary for efficient tomographyprotocols and therefore we evolved Lundeen’s proposals [4] to exact formulas that donot require the weak approximation (Section 3.2).

We compared the results with the aid of simulations (Section 3.3) and proved that theuse of strong couplings allows to obtain more precise estimates with lesser statistics,because of the sin−2(θ) law followed by the error, with θ representing the strength. Themeasured matrix elements are also more resistant to experimental systematic biasesbecause they are extracted from linear combinations of counts that are proportional tosin2(θ) and hence are more vulnerable to small errors in the data when θ → 0. We alsoargued that our methods may be preferable to standard QST when the dimension d ofthe system is large. This is because only d + 1 different projectors are needed (insteadof d2) and all but one of them correspond to states in the measurement basis (Section3.4).

The bulk of this work has been the realization of a measurement setup that can applyall these reconstruction strategies to the polarization state of light. We employed anentangled photon source (Section 4.1) to work in true quantum regime and easily gener-ate states of arbitrary purity. We first optimized the apparatus for maximum couplingstrength (Section 4.2) and then evolved it to investigate the weak regime (Section 4.3).The experimental results confirm the validity of our proposals and their ability to pro-duce estimates that are close to those of QST (trace distance . 0.15). We also showedthe predictable failure of methods based on the weak approximation to be accurate athigh values of strength (Figure 5.4, for θ = π

2 results are mostly non-physical) and thevulnerability of most protocols to experimental errors at low θ (Figure 5.12).

In conclusion, this experiment can demonstrate that our evolutions to direct state re-construction schemes are functional and represent a significant step forward in terms ofquality of the results. We believe that equivalent transitions from weak to strong mea-surement regime might provide similar improvements also in other fields of applicationof the weak value, such as the study of incompatible observables (Section 3.5).

65

Appendix A

Proof of the Theoretical Results

This appendix has the purpose of proving the most important relations that are used inthe main text. Although some of these results are already known, it is not easy to findan exhaustive and detailed derivation of all of them, therefore I hope the reader will findthe following dissertation an useful supplement to the reading of Chapters 2 and 3.

The next section will focus on equation (2.9) that shows how to extract the weak valuesfrom pointer averages, whereas the subsequent one will derive the properties of weakmeasurements of products of more observables, thus justifying formula (2.31). SectionA.3 will instead delve into mostly uncharted territory, as it will demonstrate the funda-mental relations of the new measurement protocols that are the backbone of this thesis,while also confirming Lundeen’s results in the weak limit. In Section A.4, I will showhow I extended the procedure for the measurement of products of incompatible observ-ables using the ancilla scheme to the case of arbitrary strength, thus proving equations(3.27) and (3.29). Finally, Section A.5 will provide some details on the simulations thatwere used in this work.

A.1 Measuring the Weak Value

Consider the situation described in Section 2.1. Suppose |φS〉 to be the initial state ofthe object system and to prepare ad hoc a continuous uni-dimensional ancilla in whichposition and momentum operators XA and PA are naturally defined. It is convenient tochoose its initial state |IA〉 so that its representations in the xA and pA bases are realand have zero mean (see Reference [21] for other cases). An example is the gaussianstate

IA(xA) ≡ 〈xA|IA〉 = (√

2π∆x)−12 e− x2

4∆2x (A.1)

Observable S of the object space is coupled to momentum PA of the pointer with aninteraction Hamiltonian in the form

Hint(t) = g(t)S ⊗ PA (A.2)

so that the joint evolution operator is

U = e−i~∫Hint(t)dt = e−i

g~ S⊗PA (A.3)

67

68 Proof of the Theoretical Results

It is easy to see that U defines a translation of the pointer the strength of which dependson g and on the action of S on |φS〉. For example if |φS〉 =

∑j cj |sj〉 where cj ∈ C and

|sj〉 are the eigenstates of S, the initial joint state |ψ〉 = |φS〉 ⊗ |IA〉 evolves into

ψ′(xA) ≡ 〈xA|U |ψ〉 =∑j

cj〈xA|e−ig~ sj PA (|sj〉 ⊗ |IA〉) =

∑j

cjIA(xA − gsj)|sj〉 (A.4)

On this state:

〈X(g)A 〉ψ′ =

∫dxA

∑j

∑k

IA(xA − gsj)IA(xA − gsk)xA〈sj |sk〉c?jck

=

∫dxA

∑j

∑k

IA(xA − gsj)IA(xA − gsk)xAδjkc?jck

=

∫dxA

∑j

I2A(xA − gsj)xA|cj |2

=∑j

|cj |2(〈XA〉IA + gsj) =∑j

|cj |2gsj = g〈S〉φS

(A.5)

where 〈XA〉IA = 0 has been used. This proves relation (2.4). Moreover

〈(X(g)A )2〉ψ′ =

∫dxA

∑j

I2A(xA − gsj)x2

A|cj |2 =∑j

|cj |2(∆2x + g2s2

j ) = g2〈S2〉φS + ∆2x

(A.6)in which the property of gaussian integrals

∫dxAI

2A(xA − x′)x2 = x′2 + ∆2

x has beenused. Then the variance of the measurement becomes

Var(X(g)A )ψ′ = 〈(X(g)

A )2〉ψ′ − (〈X(g)A 〉ψ′)

2 = g2 Var(S)φS + ∆2x (A.7)

Apart from known constant g, the mean value of XA coincides with that of S, so thatthis model mimics the axiomatic quantum measurement. However, the variance of thedistribution of results is also made larger by that of the pointer ∆2

x.

If g is so small that ∆x � g|sj−sk|, it is possible to approximate the evolution operatoras

U ≈ 1− i g~S ⊗ PA (A.8)

then|ψ′〉 = U |ψ〉 ≈ |φS〉 ⊗ |IA〉 − i

g

~S|φS〉 ⊗ PA|IA〉 (A.9)

On this state, neglecting terms of order higher than 1 in g:

〈X(g)A 〉ψ′ ≈ 〈φS |φS〉 ⊗ 〈XA〉IA + i

g

~〈S〉φS 〈PAXA − XAPA〉IA = g〈S〉φS

〈(X(g)A )2〉ψ′ ≈ ∆2

x + ig

~〈S〉φS 〈PAX

2A − X2

APA〉IA = ∆2x + 2g〈S〉φS 〈XA〉IA = ∆2

x

Var(X(g)A )ψ′ ≈ ∆2

x − (g〈S〉φS )2 ≈ ∆2x

(A.10)

which are first order approximations of the previous results.

Proof of the Theoretical Results 69

The weak value naturally arises if the object system is projected onto state |ΦS〉 (post-selection). Due to entanglement the joint state collapses onto

|ψpost〉 = |ΦS〉〈ΦS |ψ′〉 ≈ |ΦS〉 ⊗(〈ΦS |φS〉|IA〉 − i

g

~〈ΦS |S|φS〉PA|IA〉

)(A.11)

the norm of which is

|〈ψpost|ψpost〉| =

√|〈ΦS |φS〉|2 +

g2|〈ΦS |S|φS〉|24∆2

x

≈ |〈ΦS |φS〉| (A.12)

if it is imposed that

g2

4∆2x

∣∣∣∣∣〈ΦS |S|φS〉〈ΦS |φS〉

∣∣∣∣∣2

� 1 (A.13)

Using this normalization and ignoring the global phase of 〈ΦS |φS〉, the pointer state canbe written as

|FΦ,A〉 =〈ΦS |ψ′〉〈ΦS |φS〉

≈ |IA〉 − ig

~〈ΦS |S|φS〉〈ΦS |φS〉

PA|IA〉 (A.14)

where the weak value 〈SW 〉Φφ ≡〈ΦS |S|φS〉〈ΦS |φS〉 appears. Ancilla observables can be measured

on this state, then:

〈XA〉FΦ,A≈ i g

~<(〈SW 〉Φφ )〈PAXA − XAPA〉IA +

g

~=(〈SW 〉Φφ )〈PAXA + XAPA〉IA

= g<(〈SW 〉Φφ ) +g

~=(〈SW 〉Φφ )〈2XAPA − i~〉IA

= g<(〈SW 〉Φφ )

(A.15)

where 〈2XAPA〉IA = i~ has been used. Moreover:

〈PA〉FΦ,A≈ i g

~<(〈SW 〉Φφ )〈P 2

A − P 2A〉IA +

g

~=(〈SW 〉Φφ )〈P 2

A + P 2A〉IA =

g~2∆2

x

=(〈SW 〉Φφ )

(A.16)

where 〈P 2A〉IA = ~2

4∆2x

has been used. These two results prove relation (2.9) of the main

text. Finally

〈X2A〉FΦ,A

≈ 〈X2A〉IA = ∆2

x 〈P 2A〉FΦ,A

≈ 〈P 2A〉IA =

~2

4∆2x

(A.17)

I now show that similar expression are valid if the ancilla is a qubit. Let the initialpointer state be the eigenstate |0A〉 of σzA and let the unitary evolution be

U = e−iθS⊗σyA ≈ 1S ⊗ 1A − iθS ⊗ σyA (A.18)

where small angle θ plays the role of g. The joint state after postselection is

|ψpost〉 ≈ |ΦS〉 ⊗(〈ΦS |φS〉|0A〉+ θ〈ΦS |S|φS〉|1A〉

)(A.19)

so that, assuming

θ2

∣∣∣∣∣〈ΦS |S|φS〉〈ΦS |φS〉

∣∣∣∣∣2

� 1 (A.20)


the pointer state can be expressed as

|FΦ,A〉 ≈ |0A〉+ θ〈SW 〉Φφ |1A〉 (A.21)

Therefore〈σxA〉FΦ,A

= 〈σxA〉0A+ θ<(〈SW 〉Φφ )(〈1A|σxA|0A〉+ 〈0A|σxA|1A〉)

+ iθ=(〈SW 〉Φφ )(〈0A|σxA|1A〉 − 〈1A|σxA|0A〉)

= 2θ<(〈SW 〉Φφ )

(A.22)

and〈σyA〉FΦ,A

= 〈σyA〉0A+ θ<(〈SW 〉Φφ )(〈1A|σyA|0A〉+ 〈0A|σyA|1A〉)

+ iθ=(〈SW 〉Φφ )(〈0A|σyA|1A〉 − 〈1A|σyA|0A〉)

= 2θ=(〈SW 〉Φφ )

(A.23)

where 〈σxA〉0A = 〈σyA〉0A = 0, σxA|0A〉 = |1A〉, σxA|1A〉 = |0A〉, σyA|0A〉 = i|1A〉 andσyA|1A〉 = −i|0A〉 have all been used. Finally, 〈σ2

xA〉FΦ,A= 〈σ2

yA〉FΦ,A= 〈σ2

xA〉0A =

〈σ2yA〉0A = 1. Notice that this formalism is much simpler compared to that of the

continuous pointer.

A.2 Weak Measurements of Products

Sometimes it may be interesting to find the weak value of the product of two or moreobservables. If they commute with one another, this product is hermitian and thereforean observable itself, so that it plays the role of S in the previous formalism and it canbe treated the usual way as explained in Sections 2.1 and A.1. However, it may bemore difficult to experimentally implement the product operator compared to the singlefactors, or they may not commute, like Πaj and Πb0 in the direct weak reconstructionprotocol proposed by Lundeen. For these cases there exists a different formulation thattreats the various factors separately [45, 59].

I have shown in the previous section that

<(〈SW 〉Φφ

)≈ 1

g〈XA〉FΦ,A

=(〈SW 〉Φφ

)≈ 2∆2

x

~g〈PA〉FΦ,A

(A.24)

I define the pointer operator ΣA ≡ 12∆x

XA + i∆x~ PA so that

〈SW 〉Φφ =2∆x

g〈FΦ,A|ΣA|FΦ,A〉 (A.25)

Suppose that the initial state of the ancilla is the usual gaussian function of xA butthis time I call it |0A〉 because of its resemblance to the ground state of an harmonicoscillator. It is also evident from its definition that ΣA behaves as a lowering operatorand ΣA|0A〉 = 0, while its transpose conjugate Σ†A ≡

12∆x

XA − i∆x~ PA is a raising

operator and Σ†A|0A〉 = |1A〉. Notice moreover that PA = i ~2∆x

(Σ†A − ΣA

).


Suppose that one wants to find the weak value of

S =N∏j=1

Sj (A.26)

which is not necessarily hermitian. They can couple each observable to a separate ancillaso that the interaction Hamiltonian is

Hint = i

(~g(t)

2∆x

)∑j

Sj

(Σ†j − Σj

)(A.27)

where for simplicity the strength is assumed to be the same for all couplings, that is,gj(t) = g(t) ∀j. The joint initial state is

|ψ〉 = |φS〉 ⊗ |0〉⊗N (A.28)

where |0〉⊗N is the separable products of N pointer ground states. The evolution oper-ator is

U = e−i~∫Hintdt =

∞∑k=0

1

k!

(− i~

∫Hintdt

)k

=∑k

1

k!

g

2∆x

∑j

Sj(Σ†j − Σj)

k

= 1 +g

2∆x

∑j

Sj

(Σ†j − Σj

)+ . . .

(A.29)

Consider the term of this expansion corresponding to k = N , that is

1

N !

g

2∆x

∑j

Sj

(Σ†j − Σj

)N

(A.30)

It is the first that contains operators that can raise all the N pointers to the first excitedstate. In particular the part that does this is

1

N !

(g

2∆x

)N℘(SlΣ

†l

)(A.31)

where ℘(SlΣ

†l

)labels the sum of all the N ! possible orderings of products of the oper-

ators{SlΣ

†l | l = 1..N

}.

When U is applied to |ψ〉 and the object system is postselected on |ΦS〉, the finalnormalized pointer state is in the form

|FΦ,N 〉 = |0〉⊗N +g

2∆x

∑j

〈ΦS |S|φS〉〈ΦS |φS〉

⊗ |1j〉+ . . .

+

(g

2∆x

)N 1

N !

〈ΦS |℘(Sl)|φS〉〈ΦS |φS〉

⊗ |1〉⊗N + . . .

(A.32)


where |1j〉 is the state in which all pointers are in the ground state except for the j-thone, that is in the first excited level, whereas |1〉⊗N labels the configuration in which allthe pointers are excited. Now consider the object⟨∏

j

Σj

⟩FΦ,N

≡ 〈FΦ,N |∏j

Σj |FΦ,N 〉 (A.33)

When applying operator∏j Σj to |FΦ,N 〉 all the terms that contain a pointer still in the

ground state are annihilated:

∏j

Σj |FΦ,N 〉 =

(g

2∆x

)N 1

N !〈℘(Sj)

W 〉Φφ |0〉⊗N + . . . (A.34)

If addends of order higher than N in g are neglected⟨∏j

Σj

⟩FΦ,N

≈(

g

2∆x

)N 1

N !〈℘(Sj)

W 〉Φφ (A.35)

The term 1N !〈℘(Sj)

W 〉Φφ is the so-called joint weak value of observables {Sj | j = 1..N}.If they commute with one another, one can write

⟨∏j

Sj

W⟩Φ

φ

=1

N !〈℘(Sj)

W 〉Φφ ≈(

2∆x

g

)N ⟨∏j

Σj

⟩FΦ,N

(A.36)

In order to measure 〈∏j Σj〉FΦ,N

, which is not the mean value of an observable because

the Σjs are not hermitian, one has to expand the lowering operators in products of Xj

and Pk and measure them on different subensembles of pointers. For instance for N = 2:

〈(S1S2)W 〉Φφ =

(2∆x

g

)2

〈Σ1Σ2〉FΦ,AB

=

(2∆x

g

)2⟨( 1

2∆xX1 + i

∆x

~P1

)(1

2∆xX2 + i

∆x

~P2

)⟩FΦ,AB

(A.37)

therefore

<(〈(S1S2)W 〉Φφ

)=

(1

g

)2(〈X1X2〉FΦ,AB

− 4∆4x

~2〈P1P2〉FΦ,AB

)=(〈(S1S2)W 〉Φφ

)=

2∆2x

~

(1

g

)2 (〈X1P2〉FΦ,AB

+ 〈P1X2〉FΦ,AB

) (A.38)

If the Sjs do not commute, these relations are still valid provided one specific order ischosen. In sequential weak measurements [60], the couplings between object and ancillaeare done one after another so that the evolution operator is

U =N∏j=1

eg

2∆xSj(Σ

†j−Σj) =

∏j

∞∑k=0

1

k!

(g

2∆xSj(Σ

†j − Σj)

)k=∏j

∑k

Bkj

k!(A.39)


where for ease of notation I have defined Bkj ≡

(g

2∆xSj(Σ

†j − Σj)

)k. Just as above the

joint system evolves according to this unitary, the object is postselected on |ΦS〉 andfinally 〈

∏j Σj〉FΦ,N

is measured on the ancillae. Terms in |FΦ,N 〉 that have evolved from

a B0j contribute nothing to the result because of the action of the lowering operators on

the ground state. The first non-zero term comes from∏j B

1j and in particular contains(

g

2∆x

)N∏j

SjΣ†j (A.40)

which raises all the pointers to the first excited state. Following the above passagesagain ⟨∏

j

Sj

W⟩Φ

φ

≈(

2∆x

g

)N ⟨∏j

Σj

⟩FΦ,N

(A.41)

This strategy is used in the direct weak reconstruction of the density operator where thegoal is 〈Πb0ΠW

aj 〉aj% . The two couplings are actualized sequentially: first that of Πaj and

then one for Πb0 . In general this is the only practically feasible procedure, if the jointweak value 1

N !〈℘(Sj)W 〉Φφ is needed, one either measures all the N ! possible sequential

weak values with different orders and averages them, or uses Trotter’s formula [22, 51]

eA+B = limn→∞

(eAn e

Bn

)n(A.42)

so that a long series of alternated couplings can approximate the joint result.

In order to get closer to the formalism of the main text I now show that it easy toreplicate this discussion in the case of qubit pointers. If the coupled operator of eachancilla is σy and the initial state is the eigenstate |0〉 of σz, then one can define theladder operators:

Σ ≡ σx + iσy2

Σ†≡ σx − iσy2

(A.43)

so that Σ|0〉 = 0 and Σ†|0〉 = |1〉. If θ denotes the coupling strength, the joint weakvalue is

1

N !〈℘(Sj)

W 〉Φφ ≈1

θN

⟨∏j

Σj

⟩FΦ,N

(A.44)

and the sequential weak value for couplings made in a specific order is

⟨∏j

Sj

W⟩Φ

φ

≈ 1

θN

⟨∏j

Σj

⟩FΦ,N

(A.45)

which coincides with equation (2.31) for N = 3.


A.3 Derivation of the Measurement Protocols

I shall now prove the relations that are used in the measurement protocols of Sections2.5 and 3.2. I will focus initially on the DRDO scheme which is the most comprehensive.

Consider a d-dimensional Hilbert space HS and suppose that a system is prepared in anunknown state described by density operator %S . The goal of the following dissertationis to find the d2 matrix elements %jk = 〈aj |%S |ak〉 expressed in the orthonormal basis{|aj〉 | j = 1..d} of HS . Two bidimensional ancillae HA,B are prepared in the pure state|0〉〈0| so that the initial joint state can be written as

%tot = %S ⊗ |0A〉〈0A| ⊗ |0B〉〈0B| (A.46)

Since the pointers are built ad-hoc, it is not necessary to consider more general cases suchas multi-dimensional Hilbert spaces or mixed initial states (although purity is always anapproximation in practice) [24, 25]. For each choice of j, observables Πaj ≡ |aj〉〈aj | andσyA are coupled so that the evolution operator is

UA(aj) = e−iθAΠaj⊗σyA = (1 + (cA − 1)Πaj )⊗ 1A − isAΠaj ⊗ σyA (A.47)

which does not act on HB. (From now on I will use symbols cA,B, sA,B and tA,B in

place of cos(θA.B), sin(θA,B) and tan(θA,B

2

)). Following the abstract working principle

of any measuring device, this coupling is implemented using tools that respond to thej-th component of %S with a change of the state of the ancilla.

Subsequently a similar coupling is made between Πb0 = |b0〉〈b0| and σyB, where |b0〉 ≡1√d

∑dj=1 |aj〉. In this case the evolution operator is

UB = e−iθBΠb0⊗σyB = (1 + (cB − 1)Πb0)⊗ 1B − isBΠb0 ⊗ σyB (A.48)

which does not act on HA. By defining the operators on HS :

α0 ≡ 1 + (cA − 1)Πaj

α1 ≡ sAΠaj

β0 ≡ 1 + (cB − 1)Πb0

β1 ≡ sBΠb0

(A.49)

one can write

UA(aj) =

1∑n=0

(−i)nαnσnyA UB=

1∑m=0

(−i)mβmσmyB (A.50)

Then the evolved state is

%′tot(aj) ≡ UBUA(aj)%U†A(aj)U

†B

=∑

n,n′,m,m′

in′+m′−n−mβmαn%Sαn′ β

′m ⊗ σnyA|0A〉〈0A|σn

′yA ⊗ σmyB|0B〉〈0B|σm

′yB

(A.51)


Considering that r(jk)µν = Tr(〈ak|%′tot(aj)|ak〉⊗σ

(aj)µA ⊗σ

(b0)νB ) = Tr(Πak %S)〈σ(aj)

µA σ(b0)νB 〉Fak,AB ,

one finds:

〈Π(aj)1A Π

(b0)1B 〉Fak,AB Tr(Πak %S) = r

(jk)00 − r

(jk)30 − r

(jk)03 + r

(jk)33 =

s2As

2B

d2%jj ∀k

〈σ(aj)yA σ

(b0)yB 〉Fak,AB Tr(Πak %S) = r

(jk)22 = −2sAsB

d<(%jk) j 6= k

〈σ(aj)xA σ

(b0)yB 〉Fak,AB Tr(Πak %S) = r

(jk)12 =

2sAsBd=(%jk) j 6= k

(A.62)as stated in equation 3.7.

If θA and θB are small and terms of order higher than 1 in θAθB are neglected, then alsoequation 2.33 is verified ∀j, k:(

〈σ(aj)xA σ


(aj)yA σ

(b0)yB 〉Fak,AB

)Tr(Πak %S) = r

(jk)11 − r

(jk)22

= 4sAsB

(1− δjk)d

<(%jk) +cB − 1

d2

∑l 6=j<(%jl) + δjk

cAd<(%jk) +

cA(cB − 1)

d2%jj

≈ 4θAθB

d<(%jk)(

〈σ(aj)xA σ

(b0)yB 〉Fak,AB + 〈σ(aj)

yA σ(b0)xB 〉Fak,AB

)Tr(Πak %S) = r

(jk)12 + r

(jk)21

= 4sAsB

(1− δjk)d

=(%jk) +cB − 1

d2

∑l 6=j=(%jl)

≈ 4θAθB

d=(%jk)

(A.63)

The exact equation, along with

〈Π(aj)1A σ

(b0)xB 〉Fak,AB Tr(Πak %S) = r

(jk)01 − r

(jk)31 = δjk

2s2AsBd

%jk +2s2AsB(cB − 1)

d2%jj

〈σ(aj)xA Π


(jk)10 − r

(jk)13 =

2sAs2B

d2

∑l

<(%jl) +2sAs

2B(cA − 1)

d2%jj

〈σ(aj)yA Π


(jk)20 − r

(jk)23 =

2sAs2B

d2

∑l

=(%jl)

(A.64)


also proves formula 3.6. Indeed:(〈σ(aj)xA σ


(aj)yA σ

(b0)yB 〉Fak,AB + 2tB〈σ

(aj)xA Π

(b0)1B 〉Fak,AB + 2tA〈Π

(aj)1A σ

(b0)xB 〉Fak,AB+

+ 4tAtB〈Π(aj)1A Π

(b0)1B 〉j,k

)Tr(Πak %S)

= r(jk)11 + r

(jk)22 + 2tB

(r

(jk)10 − r

(jk)13

)+ 2tA

(r

(jk)01 − r

(jk)31

)+

+ 4tAtB

(r

(jk)00 − r

(jk)30 − r

(jk)03 + r

(jk)33

)=

4sAsBd<(%jk)(

〈σ(aj)xA σ

(b0)yB 〉Fak,AB + 〈σ(aj)

yA σ(b0)xB 〉Fak,AB + 2tB〈σ

(aj)yA Π

(b0)1B 〉Fak,AB

)Tr(Πak %S)

= r(jk)12 + r

(jk)21 + 2tB

(r

(jk)20 − r

(jk)23

)=

4sAsBd=(%jk)

(A.65)

Inverting this shows that blindly using the approximation of formula (A.63) to find thematrix elements when θA and/or θB are not small enough causes a bias

<(%Wjk − %jk) =d

4sAsB·

((r

(jk)11 + r

(jk)22

)−(r

(jk)11 + r

(jk)22 + 2tB

(r

(jk)10 − r

(jk)13

)+ 2tA

(r

(jk)01 − r

(jk)31

)+

+ 4tAtB

(r

(jk)00 − r

(jk)30 − r

(jk)03 + r

(jk)33

)))

= δjk(cA − 1)%jk +cB − 1

d

∑l

<(%jl) +(cA − 1)(cB − 1)

d%jj

=(%Wjk − %jk) =d

4sAsB·((r

(jk)12 + r

(jk)21

)−(r

(jk)12 + r

(jk)21 + 2tB

(r

(jk)20 − r

(jk)23

)))=cB − 1

d

∑l

=(%jl)

(A.66)which validates equation (3.14).

This formalism is also helpful to prove relation (3.18) of the main text. Supposing thatthe measurement results are only affected by Poissonian error, the variance associatedto the counts that one measures in order to find the mean value of projector Πµa ⊗ Πνb

is

δN2µa,νb,jk = N〈Π(aj)

µa Π(b0)νb 〉Fak,AB Tr(Πak %S) =

N

4

(r

(jk)00 + ar

(jk)µ0 + br

(jk)0ν + abr(jk)

µν

)(A.67)

where N is the total power of the signal, Πµa = 12 (1A + aσµA), Πνb = 1

2 (1B + bσνB) anda, b = ±1. As in Subsection 3.3.2, the error of N is not considered because it dependson how it is calculated or measured. However, it usually does not change the order of


magnitude of the result. The variance of observable σµA ⊗ σνB becomes

δ2µν,jk =

∑a,b

δN2µa,νb,jk

N2=rjk00

N=

Tr(〈ak|%′tot(aj)|ak〉)N

(A.68)

where

r(jk)00 = %kk +

2(cB − 1)

d

(∑l

<(%lk)−1

d

∑ll′

%ll′)

)+

+2(cA − 1)(cB − 1)

d

(<(%jk) +

(δjk −

2

d

)∑l

<(%jl) + 2

(1

d− δjk

)%jj

) (A.69)

From this expression, and also from the fact that r(jk)00 is the probability of successful

postselection on state |ak〉 after the two evolutions, it is clear that∑

k r(jk)00 = 1, which

demonstrates formula (3.19).

The proof of the DRDD protocol is easier because only one coupling is involved. Similarlyto the previous discussion, the starting point is

%tot = %S ⊗ |0A〉〈0A| (A.70)

and the only evolution operator is

UA(aj) = e−iθAΠaj⊗σyA = (1+(cA−1)Πaj )⊗1A−isAΠaj⊗σyA =1∑

n=0

(−i)nαnσnyA (A.71)

again with

α0 ≡ 1 + (cA − 1)Πaj α1≡ sAΠaj (A.72)

The result of the evolution is

%′tot(aj) ≡ UA(aj)%U†A(aj) =

∑n,n′

in′−nαn%Sαn′ ⊗ σnyA|0A〉〈0A|σn

′yA (A.73)

but this time postselection occurs on |bl〉 ≡ 1√d

∑d−1m=0 |am〉e

i 2πmld with l ∈ {0..d− 1}.

The final state is the separable product of |bl〉〈bl| in the object system and

〈bl|%′tot(aj)|bl〉 =∑n,n′

in′−n〈bl|αn%Sαn′ |bl〉 ⊗ σnyA|0A〉〈0A|σn

′yA =

1

2

4∑µ=0

r(jl)µ σµA (A.74)

for the ancilla. In this formalism

r(jl)µ ≡ Tr(〈bl|%′tot(aj)|bl〉 ⊗ σµA)

=∑n,n′

in′−n〈bl|αn%Sαn′ |bl〉 · 〈0A|σn

′yAσµAσ

nyA|0A〉

=∑n,n′

in′−n〈bl|αn%Sαn′ |bl〉 ·Gµn,n′

(A.75)


and just as above

Gµn,n′ ≡ δµ,0δn,n′ + δµ,1i2n−1δn,1−n′ + δµ,2δn,1−n′ + δµ,3(−1)nδn,n′ (A.76)

Considering that

α0|bl〉 =1√d

d−1∑m=0

ei2πmld |am〉+

cA − 1√d

ei2πjld |aj〉 α1|bl〉=

sA√dei

2πjld |aj〉 (A.77)

one can write:

r(jl)0 = 〈bl|α0%Sα0|bl〉+ 〈bl|α1%Sα1|bl〉

=1

d

∑m,m′

ei2π(m′−m)l

d %mm′ +2(cA − 1)

d

∑m

<(ei

2π(m−j)ld %jm

)− 2(cA − 1)

d%jj

r(jl)1 = 〈bl|α0%Sα1|bl〉+ 〈bl|α1%Sα0|bl〉

=2sAd

∑m

<(ei

2π(m−j)ld %jm

)+

2sA(cA − 1)

d%jj

r(jl)2 = i〈bl|α0%Sα1|bl〉 − i〈bl|α1%Sα0|bl〉

=2sAd

∑m

=(ei

2π(m−j)ld %jm

)r

(jl)3 = 〈bl|α0%Sα0|bl〉 − 〈bl|α1%Sα1|bl〉

=1

d

∑m,m′

ei2π(m′−m)l

d %mm′ +2(cA − 1)

d

∑m

<(ei

2π(m−j)ld %jm

)+

2cA(cA − 1)

d%jj

(A.78)

According to formula (3.9)

ρA1(jl) =1

2sATr(Πbl %S)(〈σ(aj)

xA 〉Fbl,A + i〈σ(aj)yA 〉Fbl,A) =

1

2sA

(r

(jl)1 + ir

(jl)2

)=

1

d

∑m

ei2π(m−j)l

d %jm +cA − 1

d%jj

ρA2(jl) =1

sATr(Πbl %S)〈Π(aj)

1A 〉Fbl,A =1

2sA

(r

(jl)0 − ir(jl)

3

)=sAd%jj

(A.79)

so that Djl = ρA1(jl) + tAρA2(jl) = 1d

∑m e

i2π(m−j)l

d %jm coincides with the jl-th element

of the discrete Dirac distribution. Moreover, due to the fact that 1d

∑l ei2π(a−b)l

d = δab,one has

d · tAδjkρA2(jl) +∑l

e2πil(j−k)

d ρA1(jl)

= (1− cA)δjk%jj +∑m

1

d

∑l

ei2π(m−k)l

d %jm +cA − 1

d

∑l

ei2π(j−k)l

d %jj

= %jk

(A.80)

which validates this protocol as a way to find the density operator. When θA,B is small,one can neglect the contribution of ρA2(jl) and find Lundeen’s proposal of (2.35).


A.4 Strong Measurements of Products

I now explain how to measure the mean value of an operator on the object system HS

that can be written as∏Nj=1 Πj,S , using the ancilla scheme and without any approxima-

tion. Initially, the system is prepared in

%S ⊗ (|0〉〈0|)⊗N (A.81)

in which the superscript ⊗N labels the tensor product over all the N pointers. Then, thevarious projectors Πj are coupled sequentially and in reverse order to the σy observablesof each ancilla so that the evolution operator of the joint system is

U =N∏j=1

Uj = U1..UN (A.82)

where each Uj acts only on HS and on the j-th meter as

Uj = e−iθjΠj,S⊗σy,j = (1S + (sj − 1)Πj,S)⊗ 1j − isAΠj,S ⊗ σy,j (A.83)

(again sj ≡ sin(θj), cj ≡ cos(θj) and tj ≡ tan(θj2

)).

The goal of this proof is to show that, as in relation (3.27), it is possible to define

Ej ≡σx,j + iσy,j

2+ tjΠ1,j (A.84)

and then to write ⟨N∏j=1

Ej

⟩FN

=N∏j=1

sj

⟨N∏j=1

Πj,S

⟩%S

(A.85)

First, notice the left-hand side can be expressed as⟨N∏j=1

Ej

⟩FN

= Tr(1S ⊗ E⊗N U1..UN %S ⊗ (|0〉〈0|)⊗N U †N ..U

†1

)= TrS

(〈0|⊗N U †N ..U

†11S ⊗ E

⊗N U1..UN |0〉⊗N %S) (A.86)

so that (A.85) is equivalent to

〈0|⊗N U †N ..U†11S ⊗ E

⊗N U1..UN |0〉⊗N =N∏j=1

sjΠj,S (A.87)

More generally, consider

〈0|⊗N U †N ..U†1AS ⊗ E

⊗N U1..UN |0〉⊗N = AS

N∏j=1

sjΠj,S (A.88)

where AS is an arbitrary operator on HS . It is easy to prove that this expressionis valid for N = 1. Indeed since U1|01〉 = (1S + (c1 − 1)Π1,S)|01〉 + s1Π1,S |11〉 and


E1U1|01〉 = s1Π1,S(|01〉+ t1|11〉), one arrives at

〈01|U †1AS ⊗ E1U1|01〉 = s1ASΠ1,S + s1(c1 − 1)Π1,SASΠS + s21t1Π1,SASΠS

= s1ASΠj,S

(A.89)

Now suppose that (A.88) holds true for a given N , then it must also be valid for N + 1.Indeed

〈0|⊗N+1U †N+1..U†1AS ⊗ E

⊗N+1U1..UN+1|0〉⊗N+1

= 〈0N+1|U †N+1〈0|⊗N U †N ..U

†1AS ⊗ E

⊗N U1..UN |0〉⊗N ⊗ EN+1UN+1|0N+1〉

= 〈0N+1|U †N+1

AS N∏j=1

sjΠj,S

⊗ EN+1UN+1|0N+1〉

(A.90)

Now, defining AS∏Nj=1 sjΠj,S = A′S one can repeat the passages of (A.89) and get

〈0N+1|U †N+1A′S ⊗ EN+1UN+1|0N+1〉 = sN+1A

′SΠN+1,S (A.91)

so that in conclusion

〈0|⊗N+1U †N+1..U†1AS ⊗ E

⊗N+1U1..UN+1|0〉⊗N+1 = AS

N+1∏j=1

sjΠj,S (A.92)

Since (A.88) is true for N = 1 and if it is for an N , it is also for N + 1, it is necessarilyvalid for any finite N . In particular, when AS = 1S , relation (3.27) is verified, but onecan also choose AS = Π0,S , thus adding an additional projector to the products. In this

way it is possible to measure 〈∏Nj=0 Πj,S〉 using N couplings and one postselection on

the object system.

I now show that it is possible to ignore the contribution of one of the projectors, forinstance Πl, by measuring E′l ≡ Π0,l + tlσx,l + t2l Π1,l instead of El on the associated

pointer. Indeed, since E′lUl|0l〉 = |0l〉+ tl|1l〉, simply

〈0l|U †l AS ⊗ E′lUl|0l〉 = AS (A.93)

This means that one can let the system evolve with the same unitary transformationbut can choose to ignore one (or more) of the contributions with a simple change ofthe pointer measurements. Any subsequent observation can either concatenate a newsjΠj,S term to the previous AS (when Ej is measured) or leave it as it is (when E′j ismeasured). In conclusion a generalization of equation (3.29) is⟨∏

j∈Λ

Ej∏j′∈Λ′

E′j′

⟩=∏j∈Λ

sj

⟨∏j∈Λ

Πj,S

⟩(A.94)

Notice that at first order in small strength, E ≈ Σ (Section A.2), whereas 〈0l|U †l AS ⊗E′lUl|0l〉 ≈ 〈0l|U

†l AS ⊗ Π0,lUl|0l〉 ≈ 〈0l|U †l ASUl|0l〉 ≈ AS , thus confirming the methods

used in References [50, 51]. At maximum strength, instead, E′ = 1 + σx = 2Π+ can bemeasured with a single acquisition, as it is a multiple of a pointer projector.


A.5 Details about the Simulations

A.5.1 Generation of Random Density Matrices

In Section 3.3, the results of the various protocols are compared in different conditionsof coupling strength. In order to access a large pool of initial states we randomlygenerated 104 bidimensional density operators and for each we calculated the predictionsof Lundeen’s DRDO method using formula (3.12) and the analytical relations reportedin (A.61). We obtained the average, minimum and maximum trace distances betweenthese results and the actual density matrices for 200 different settings of strength (Figure3.1a) as well as the frequency of distances higher than 0.1 (Figure 3.1b). We also foundthe analytical results of formulas (3.18) and (3.20) (neglecting the contribution of 1√

N) in

order to estimate the standard deviation associated to each pointer mean value (Figure3.2a) and the relative error on a matrix element (Figure 3.2b).

The only step which involves a random component is the initial generation of the truestates. We followed the recipe introduced in References [61, 62] which is based on solidphysical grounds. First, we imagined to couple the bidimensional (d = 2) Hilbert spaceH to another qubit system Haux. We generated 8 real numbers according to a normaldistribution with zero mean and unit variance. They are the real and imaginary parts ofthe 4 coefficients that identify pure state |ψ〉 ∈H ⊗Haux expressed in a product basis:

|ψ〉 =d=2∑j=1

daux=2∑k=1

cjk|aj〉 ⊗ |bk〉 (A.95)

We arranged these cjk in a d× daux = 2× 2 matrix X so that Xjk = cjk which is calleda Ginibre matrix. We then calculated the d× d operator

% = XX† (A.96)

which coincides with the partial trace of |ψ〉〈ψ| on system Haux. Indeed:

TrHaux (|ψ〉〈ψ|) =

d=2∑j,j′=1

daux=2∑k,k′,l=1

〈bl|cjk|ajbk〉〈aj′bk′ |c?j′k′ |bl〉 =∑

j,j′,k,k′,l

δklδk′lcjkc?j′k′ |aj〉〈aj′ |

=∑jj′k

cjkc?j′k|aj〉〈aj′ | = XX†

(A.97)Finally we normalized % with its own trace in order to end with a matrix that satisfiesall the properties of a density operator.

This procedure based on the partial trace resembles the one we used in the actual exper-iment and assures the production of general states. The fact that daux = d guaranteesthat the results are distributed according to the Hilbert-Schmidt metric, which inducesthe trace distance [63], but the choice of using qubit systems was only made to bettermatch the experiment conditions and save computational time. Figure A.1 shows thedistribution of 1000 samples in the Bloch Sphere:


|D |L

|H

Figure A.1: 1000 randomly generated states.

A.5.2 Simulation of the Experiment

A completely unrelated simulation was used in connection with the experiment in or-der to estimate the errors associated to the trace distances of Figures 5.4, 5.6, 5.9 and5.12. We generated 104 virtual repetitions of each experiment, using random counts dis-tributed according to Poisson functions with parameters (mean and variance) coincidingwith the actual measurement results. Of course this is not necessarily accurate as theoutcome of a single measure is not the mean of repeated acquisitions, but is the bestestimate with the available data. For each virtual dataset we performed the same analy-sis that we applied to the actual measurement results and extracted the trace distancesbetween each protocol and a reference (the QST estimate or the expected pure state).The standard deviations of these distances over the 104 samples appear as error bars inthe aforementioned figures.

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direct reconstruction of the quantum density operator via

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